Effect of dimensionality on the optical absorption properties of CsPbI₃ perovskite nanocrystals

Colloidal nanocrystals, following decades of extensive study, have begun maturing as a material platform for commercial applications such as displays¹ and photovoltaics.² However, despite more than 30 years of research into alternative material platforms, the initial chalcogenide-based colloidal nanocrystals have remained superior in both performance and stability for practical devices. Recently, synthesis of cesium lead-halide perovskite nanocrystals has been achieved,³ which has generated much excitement due to their exceptional optical properties.

Shortly following the initial synthesis of perovskite nanocubes (NCs), synthesis of perovskite nanoplatelets (NPs)^4,5 was also achieved to further broaden the gamut of applications for perovskite nanocrystals. Compared to their nanocube counterparts, the nanoplatelet geometry offers directional light absorption/emission⁶ as well as reduced dielectric screening (leading to greatly enhanced exciton binding energies⁷ and radiative recombination rates^8,9). Recently, these attractive properties have led to intense efforts in applying perovskite nanoplatelets toward a variety of applications such as light-emitting diodes¹⁰ and photovoltaics.¹¹ Understanding how electronic dynamics underlying the photophysics of perovskite nanocrystals change with the nanocrystal geometry is crucial for such practical applications. In particular, perovskite nanoplatelets have been seldom studied at cryogenic temperature to elucidate electron-phonon coupling in the material.

Here, we study CsPbI₃ perovskite nanocube and nanoplatelet ensembles at cryogenic temperatures. Absorption spectra reveal an anomalous bandgap shift to higher energies with increasing temperature, which we attribute to bandgap renormalization via electron-phonon coupling. A low-energy absorption tail is also observed in CsPbI₃ nanocubes that is likely due to shallow trap states, which implies that iodide perovskite nanocrystals may be less defect-tolerant than their bromide and chloride counterparts at low temperatures.

The orthorhombic perovskite lattice structure of the CsPbI₃ nanocrystals^12–14 is shown in Fig. 1(a), and transmission electron micrographs of the nanocubes are shown in Fig. 1(b). The measurement of 100 nanocubes reveals an average side length of 8.7 ± 2.6 nm. Although significant size and shape dispersion of the nanoplatelets preclude well-defined average side lengths, their lateral dimensions are on the order of tens of nanometers (see the supplementary material). Their bandgap energy then indicates the out-of-plane thickness to be primarily four polyhedral layers and above.

The nanocubes are synthesized according to the procedures detailed by Protesescu et al.,^3,15 and the nanoplatelets are synthesized via a method¹⁶ modified from that reported by Sheng et al.¹⁷ Brief descriptions of each method are detailed in the supplementary material.

To study their optical properties at cryogenic temperatures, we redisperse the nanocrystals in heptamethylnonane, a branched alkane that forms a transparent glass at cryogenic temperatures.¹⁹ The colloidal suspension is held in a custom sample holder approximately 0.5 mm thick and mounted on a cold-finger cryostat. Absorption spectra are measured with a broadband white light source and a UV-vis diode array spectrometer.

CsPbI₃ nanocube absorption spectra normalized to the lowest-energy 1S exciton absorption peak at temperatures ranging from 4 K to 140 K are plotted in Fig. 1(c). Although multiple peaks are observed that correspond to distinct exciton transitions, here we focus on the 1S exciton absorption peak that reflects the fundamental electronic bandgap (energy-gap) of the nanocrystals. As temperature increases, the bandgap exhibits a pronounced blue shift to higher energies, which is contrary to the red shift observed in most solids. In the literature, this phenomenon has been referred to as an anomalous bandgap shift.^20–23 

To quantify the bandgap shift, we fit the peaks with Gaussian line shapes that reflect the size distribution of the nanocrystals. As shown in Fig. 2(a), we fit only the top of each peak due to absorption tails present at lower temperatures. The widths σ of each Gaussian fit, allowed to vary freely, do not change significantly with temperature (mean width 41.81 meV and standard deviation 3.37 meV). The fitted Gaussian center energies (which agree closely with center energies found from a fourth-order polynomial fit) are plotted in Fig. 2(b), which reveals interesting behavior at temperatures below 50 K. Specifically, two clear inflection points at 20 and 30 K are observed that reveal more complicated bandgap behavior than previously reported for photoluminescence measurements of similar perovskite nanocubes.²³ 

The dependence of the electronic bandgap on temperature T may be expressed as^22,24

The first term E₀ is the intrinsic material bandgap at T = 0, and the coefficient A in the second term characterizes the change in the bandgap due to lattice unit cell expansion/contraction (in the so-called quasiharmonic approximation²⁴). Here, the change in quantum confinement energy due to the expansion/contraction of the nanocrystal volume, which we expect to be negligible at low temperatures,²⁵ is ignored. The third term then represents renormalization of the bandgap due to electron-phonon interactions, where n is summed over all phonon branches and all wave-vectors within the Brillouin zone for each branch. B_n and ℏω_n are the electron-phonon coupling strength and vibrational energy, respectively, for mode n. Whether B_n is positive or negative, resulting in an increase or decrease of the bandgap, respectively, arises from a complex interplay of microscopic dynamics and cannot be predicted easily from the properties of a given phonon branch.^20,26 However, accounting for all possible phonon branches throughout the Brillouin zone is often unnecessary in modeling the behavior of real systems. Instead, one²⁷ or two²⁰ vibrational modes are usually assumed dominant (referred to as one-oscillator and two-oscillator models), which reduces the summation to either one or two terms, respectively.

Here, we find both the one-oscillator and two-oscillator models to be insufficient in modeling the bandgap temperature dependence observed for CsPbI₃ nanocubes. As mentioned above, two inflection points are observed that necessitate at least three dominant vibrational modes that independently renormalize the bandgap. A least-squares fit of the bandgap temperature dependence to this three-oscillator model is plotted in Fig. 2(b), where good agreement is observed at both high and low temperatures. The fitted parameters are E₀ = 1916.9 meV, A = 0.3 meV/K, ℏω₁ = 5.38 meV, ℏω₂ = 5.91 meV, ℏω₃ = 17.02 meV, B₁ = −698.01 meV, B₂ = 821.67 meV, and B₃ = −217.39 meV. Instead of the acoustic and optical phonon categories that are usually invoked for two-oscillator models,^22,23 a three-oscillator model in perovskite materials align more naturally to the bending, stretching, and rocking perovskite vibrational modes that possess distinct ranges of vibrational energies.²⁸ 

We note that although the two-oscillator model was recently invoked by Saran et al. to model the temperature dependence of photoluminescence center energy in perovskite nanocrystals,²³ the data points taken at low temperatures (below 50 K) were too sparse to resolve the two inflection points we observe. Their resultant fitted bandgap dependence is plotted in Fig. 2(b) for comparison. In contrast to features in absorption spectra, which are simply proportional to the oscillator strength of each optical transition, features in photoluminescence spectra depend on many other temperature-dependent factors such as the equilibrium fine-structure carrier distribution^29,30 and emission Stokes shifts.³¹ It is therefore unclear whether the apparent two-oscillator behavior of their measurements on CsPbI₃ nanocubes was due to coarse-graining effects or the above confounding factors in temperature-dependent photoluminescence.

Lower-energy absorption tails are observed. For ideal nanocubes, the exciton density of states are comprised of delta functions that result in roughly Gaussian absorption peaks (reflecting the nanocrystal size distribution). Absorption tails at lower-energy are therefore indicative of corresponding tails of the electronic density of states, often attributed to impurities³² or surface states.³³ As shown in Fig. 2(a), the absorption peak is Gaussian at 140 K and develops a lower-energy tail with decreasing temperature. We attribute this tail to shallow defect states surrounding the valence band edge that have been shown to arise from lattice point defects.³⁴ At high temperatures, valence band electrons populate the band edge in a thermal equilibrium distribution. At low temperatures, those electrons then fill the defect states from lowest energy upward, which comprise a Halperin-Lax type distribution³⁵ with a exp(⁠$E$⁠) dependence.^36,37 The disappearance of the tail at 140 K thus suggests a few millielectronvolts (comparable to the 140 K Boltzmann energy of 12 meV) defect state energy distribution. Although, in principle, such defect state absorption should manifest in photoluminescence spectra as well, no clear band-tailing was observed in low-temperature photoluminescence measurements.²³ This is unsurprising since above-gap excitation results in competing band edge and defect state relaxation pathways and emission Stokes shifts (on the order of tens of millielectronvolts in perovskite nanocrystals^3,23) likely differ for defect transitions. For additional comparison, absorption measurements were also performed on CsPbBr₃ nanocubes (see the supplementary material for absorption spectra and synthesis methods). Although a large anomalous bandgap shift was observed (approximately 40 meV from 6 to 140 K), no absorption tail forms at low temperatures.

Electron-phonon coupling that renormalizes the CsPbI₃ bandgap should depend strongly on dimensionality in nanocrystals. In particular, lowering dimensionality should reduce electron-phonon coupling by restricting certain vibrational modes. To investigate the effect of lattice dimensionality on electron-phonon coupling, we repeat the same temperature-dependent absorption measurements on CsPbI₃ nanoplatelets. At room-temperature, a single nanoplatelet absorption peak is observed that is blue shifted relative to the nanocube bandgap due to strong quantum confinement in the out-of-plane direction. At cryogenic temperatures, shown in Fig. 3, the absorption spectrum changes in two surprising ways. First, the nanoplatelet absorption peak continues narrowing below 140 K (with no absorption tail), in contrast to the nanocube absorption peak width that remains constant at low temperatures. Second, an additional lower-energy peak also appears with decreasing temperature [see Fig. 3(a)], which, due to its center energy, we attribute to cosynthesized CsPbI₃ nanocubes. Temperature-dependent surface plots of both the nanocube and nanoplatelet peaks (measured from the same absorption spectra) are shown in Fig. 3(b), demonstrating the relative changes in peak optical density.

Again the nanoplatelet absorption peaks are fitted to the Gaussian line shapes, and the fitted center energies are plotted in Fig. 3(c). The nearly linear anomalous bandgap shift indicates weakened electron-phonon interactions and greater importance of bandgap renormalization due to unit cell expansion/contraction with temperature. To quantify these changes, we perform a linear fit of the center energy temperature-dependence. The fitted parameters are E₀ = 2055.4 meV and A = 0.2 meV/K, where A is comparable to its corresponding nanocube value. Therefore, decreasing dimensionality greatly reduces vibrational bandgap renormalization without strongly affecting that due to changes in unit cell size.

The fitted Gaussian widths σ, plotted in the inset in Fig. 3(c), reveal another interesting aspect of the electronic properties of perovskite nanoplatelets. While the nanocube absorption peak width is approximately constant at cryogenic temperatures, reflecting its inhomogeneously broadened nature, the much narrower nanoplatelet absorption peak exhibits a monotonic decrease in σ with decreasing temperature. This indicates that homogeneous broadening in perovskite nanoplatelets contributes even down to cryogenic temperatures. However, a plateau in the linewidth that decreases below 50 K, despite homogeneous out-of-plane confinement, reveals an intrinsic ensemble absorption linewidth between 10 and 11 meV. At such energy scales, inhomogeneous broadening due to variation in the in-plane confinement of exciton center-of-mass motion, usually considered to be negligible,³⁸ could become important. More advanced spectroscopic techniques such as multidimensional coherent spectroscopy³⁹ are needed to disentangle inhomogeneous and homogeneous broadening mechanisms in perovskite nanoplatelets.^19,40

In summary, the absorption of CsPbI₃ perovskite nanocrystals is measured at cryogenic temperatures. In conjunction with the anomalous bandgap shifts to higher energies with increasing temperature, additional inflection points are observed at low temperatures; which we attribute to bandgap renormalization by three vibrational modes in CsPbI₃ nanocubes, contrary to a recent study.²³ The measurement of CsPbI₃ nanoplatelets then reveals greatly reduced vibrational bandgap renormalization, which suggests that lowered nanocrystal dimensionality leads to weakened influence of lattice vibrations on electronic dynamics. Finally, absorption tails are found to form in CsPbI₃ nanocubes at low temperatures, which we attribute to defect states surrounding the valence band edge. While perovskite nanocrystals have been found to be exceptionally defect-tolerant,⁴¹ our finding suggests that shallow defects may begin to influence the optical properties of iodide nanocubes at cryogenic temperatures. This work motivates further study of electron-phonon coupling in perovskite nanocrystals to minimize their deleterious effects.

In the supplementary material, we provide details of the nanocrystal synthesis procedures, absorption temperature dependences of CsPbBr₃ nanocubes, and supplemental TEM images of the iodide nanoplatelets and bromide nanocubes.

This work was supported by the Department of Energy (Grant No. DE-SC0015782) and by the Sao Paulo Research Foundation (Grant No. 2013/16911-2). D.B.A. and G.N. acknowledge support from fellowships from the Brazilian National Council for Scientific and Technological Development (CNPq). This research was also supported by LNNano/CNPEM/MCTIC, where the TEM measurements were performed.