Effect of Dimensionality on the Optical Absorption Properties of CsPbI 3 Perovskite Nanocrystals
Albert Liu, Luiz G. Bonato, Francesco Sessa, Diogo B. Almeida, Erik Isele, Gabriel Nagamine, Luiz F. Zagonel, Ana F. Nogueira, Lazaro A. Padilha, Steven T. Cundiff
EEffect of Dimensionality on the Optical Absorption Properties of CsPbI Perovskite Nanocrystals
Effect of Dimensionality on the Optical Absorption Properties of CsPbI Perovskite Nanocrystals
Albert Liu, L. G. Bonato, Francesco Sessa, Diogo B. Almeida,
1, 3
Erik Isele, G. Nagamine, L. F. Zagonel, A.F. Nogueira, L. A. Padilha, a) and Steven T. Cundiff b) Department of Physics, University of Michigan, Ann Arbor, Michigan, USA Instituto de Quimica, Universidade Estadual de Campinas, Campinas, Sao Paulo, Brazil Instituto de Fisica, Universidade Estadual de Campinas, Campinas, Sao Paulo, Brazil Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan,USA (Dated: 15 October 2019)
The band-gaps of CsPbI perovskite nanocrystals are measured by absorption spectroscopy at cryogenic temperatures.Anomalous band-gap shifts are observed in CsPbI nanocubes and nanoplatelets, which are modeled accurately byband-gap renormalization due to lattice vibrational modes. We find that decreasing dimensionality of the CsPbI latticein nanoplatelets greatly reduces electron-phonon coupling, and dominant out-of-plane quantum confinement results ina homogeneously broadened absorption lineshape down to cryogenic temperatures. An absorption tail forms at low-temperatures in CsPbI nanocubes, which we attribute to shallow defect states positioned near the valence band-edge. I. INTRODUCTION
Colloidal nanocrystals, following decades of extensivestudy, have begun maturing as a material platform for com-mercial applications such as displays and photovoltaics .However, despite more than 30 years of research into alter-native material platforms, the initial chalcogenide-based col-loidal nanocrystals have remained superior in both perfor-mance and stability for practical devices. Recently, synthesisof cesium lead-halide perovskite nanocrystals was achieved ,which has generated much excitement due to their exceptionaloptical properties.Shortly following the initial synthesis of perovskitenanocubes, synthesis of perovskite nanoplatelets was alsoachieved to further broaden the gamut of applications forperovskite nanocrystals. Compared to their nanocube coun-terparts, the nanoplatelet geometry offers directional lightabsorption/emission as well as reduced dielectric screen-ing (leading to greatly enhanced exciton binding energies and radiative recombination rates ). Recently, these attrac-tive properties have led to intense efforts in applying per-ovskite nanoplatelets towards a variety of applications suchas light-emitting diodes and photovoltaics . Understandinghow electronic dynamics underlying the photo-physics of per-ovskite nanocrystals change with nanocrystal geometry is cru-cial for such practical applications. In particular, perovskitenanoplatelets have been seldom studied at cryogenic temper-ature to elucidate electron-phonon coupling in the material.Here, we study CsPbI perovskite nanocube andnanoplatelet ensembles at cryogenic temperatures. Ab-sorption spectra reveal an anomalous band-gap shift to higherenergies with increasing temperature, which we attributeto band-gap renormalization via electron-phonon coupling.A low-energy absorption tail is also observed in CsPbI nanocubes that is likely due to shallow trap states, which a) Electronic mail: padilha@ifi.unicamp.br b) Electronic mail: [email protected] implies that iodide perovskite nanocrystals may be lessdefect-tolerant than their bromide and chloride counterpartsat low temperatures.
II. EXPERIMENT
The orthorhombic perovskite lattice structure of the CsPbI nanocrystals is shown in Fig. 1(a), and transmission elec-tron micrographs of the nanocubes are shown in Fig. 1(b).Measurement of 100 nanocubes informs an average sidelength of 8.7 ± , and the nanoplatelets aresynthesized via a method modified from that reported bySheng, et al. . Brief descriptions of each method are detailedin the Supplemental Information.To study their optical properties at cryogenic tempera-tures we redisperse the nanocrystals in heptamethylnonane,a branched alkane that forms a transparent glass at cryogenictemperatures . The colloidal suspension is held in a customsample holder approximately 0.5 mm thick and mounted in acold-finger cryostat. Absorption spectra are measured with abroadband white light source and a UV-vis diode array spec-trometer. III. RESULTS AND DISCUSSSION
CsPbI nanocube absorption spectra normalized to thelowest-energy 1S exciton absorption peak at temperaturesranging from 4 K to 140 K are plotted in Fig. 1(c). Althoughmultiple peaks are observed that correspond to distinct excitontransitions, here we focus on the 1S exciton absorption peak a r X i v : . [ phy s i c s . a pp - ph ] O c t ffect of Dimensionality on the Optical Absorption Properties of CsPbI Perovskite Nanocrystals 2 Cs Pb I A b s o r p t i o n ( a . u . )
4K 140K (a)(b) (c) a c b a
FIG. 1. (a) Two perspective views of the orthorhombic perovskite lattice structure of CsPbI with axes as shown (plotted using the VESTAsoftware ). The unit cell is denoted by the solid black lines. (b) Transmission electron micrograph of nanocubes. (c) Absorption spectra ofCsPbI nanocrystals at temperatures ranging from 4 K to 140 K as indicated. The full-range spectra are plotted inset, while the 1S excitonpeak outlined by the dashed box is shown in the main plot. The specific temperatures plotted are indicated by the data in Fig. 2(b). that reflects the fundamental electronic band-gap (energy-gap)of the nanocrystals. As temperature increases the band-gapexhibits a pronounced blue-shift to higher energies, which iscontrary to the red-shift observed in most solids. In the liter-ature, this phenomenon has been referred to as an anomalousband-gap shift .To quantify the band-gap shift, we fit the peaks withGaussian lineshapes that reflect the size distribution of thenanocrystals. As shown in Fig. 2(a), we fit only the top ofeach peak due to absorption tails present at lower tempera-tures. The widths σ of each Gaussian fit, allowed to varyfreely, do not change significantly with temperature (meanwidth 41.81 meV and standard deviation 3.37 meV). The fit-ted Gaussian center energies (which agree closely with centerenergies found from a fourth-order polynomial fit) are plottedin Fig. 2(b), which reveals interesting behavior at tempera-tures below 50 K. Specifically, two clear inflection points at20 and 30 K are observed that reveal more complicated band-gap behavior than previously reported for photoluminescencemeasurements of similar perovskite nanocubes .The dependence of the electronic band-gap on temperature T may be expressed as : E g ( T ) = E + AT + ∑ n B n (cid:18) e ¯ h ω n / k B T − + (cid:19) . (1)The first term E is the intrinsic material band-gap at T =
0, and the coefficient A in the second term char-acterizes the change in band-gap due to lattice unitcell expansion/contraction (in the so-called quasi-harmonicapproximation ). Here the change in quantum confinement energy due to expansion/contraction of nanocrystal volume,which we expect to be negligible at low temperatures , isignored. The third term then represents renormalization ofthe band-gap due to electron-phonon interactions, where n is summed over all phonon branches and all wave-vectorswithin the Brillouin zone for each branch. B n and ¯ h ω n are theelectron-phonon coupling strength and vibrational energy re-spectively for mode n . Whether B n is positive or negative, re-sulting in an increase or decrease of the band-gap respectively,arises from a complex interplay of microscopic dynamics andcannot be predicted easily from the properties of a givenphonon branch . However, accounting for all possiblephonon branches throughout the Brillouin zone is often un-necessary in modeling the behavior of real systems. Instead,one or two vibrational modes are usually assumed domi-nant (referred to as one-oscillator and two-oscillator models)which reduces the summation to either one or two terms re-spectively.Here, we find both the one-oscillator and two-oscillatormodels to be insufficient in modeling the band-gap temper-ature dependence observed for CsPbI nanocubes. As men-tioned above, two inflection points are observed that neces-sitate at least three dominant vibrational modes that inde-pendently renormalize the band-gap. A least-squares fit ofthe band-gap temperature dependence to this three-oscillatormodel is plotted in Fig. 2(b), where good agreement is ob-served at both high and low temperatures. The fitted parame-ters are E = . A = . h ω = .
38 meV,¯ h ω = .
91 meV, ¯ h ω = .
02 meV, B = − .
01 meV, B = .
67 meV, and B = − .
39 meV. Instead of theffect of Dimensionality on the Optical Absorption Properties of CsPbI Perovskite Nanocrystals 3 (a)(b) A b s o r p t i o n ( a . u . ) FIG. 2. (a) Gaussian peak fits of CsPbI absorption spectra at threerepresentative temperatures 4, 90, and 140 K. A low-energy absorp-tion tail, indicated by the shaded gray region, forms at low temper-ature. (b) Dark-blue dots show fitted Gaussian center energy as afunction of temperature, which reflects the material band-gap. Atwo-oscillator (2-O) model using the fitted parameters from Saranet al. and a fit to the three-oscillator (3-O) model described in thetext are then plotted as the dashed black curve and solid light-bluecurve respectively. The fitted Gaussian widths σ are plotted inset. acoustic and optical phonon categories that are usually in-voked for two-oscillator models , a three-oscillator modelin perovskite materials align more naturally to the bending,stretching, and rocking perovskite vibrational modes that pos-sess distinct ranges of vibrational energies .We note that although the two-oscillator model was re-cently invoked by Saran et al. to model the temperature de-pendence of photoluminescence center energy in perovskitenanocrystals , the data points taken at low temperatures(below 50 K) were too sparse to resolve the two inflec-tion points we observe. Their resultant fitted band-gap de- pendence is plotted in Fig. 2b for comparison. In contrastto features in absorption spectra, which are simply propor-tional to the oscillator strength of each optical transition, fea-tures in photoluminescence spectra depend on many othertemperature-dependent factors such as the equilibrium fine-structure carrier distribution and emission Stokes shifts .It is therefore unclear whether the apparent two-oscillator be-havior of their measurements on CsPbI nanocubes was dueto coarse-graining effects or the above confounding factors intemperature-dependent photoluminescence.Lower-energy absorption tails are observed. For idealnanocubes, the exciton density of states are comprised of deltafunctions that result in roughly Gaussian absorption peaks (re-flecting the nanocrystal size distribution). Absorption tails atlower-energy are therefore indicative of corresponding tails ofthe electronic density of states, often attributed to impurities or surface states . As shown in Fig. 2(a), the absorptionpeak is Gaussian at 140 K and develops a lower-energy tailwith decreasing temperature. We attribute this tail to shal-low defect states surrounding the valence band-edge that havebeen shown to arise from lattice point defects . At hightemperatures valence band electrons populate the band-edgein a thermal equilibrium distribution. At low temperaturesthose electrons then fill the defect states from lowest energyupwards, which comprise a Halperin-Lax type distribution with a exp( √ E ) dependence . The disappearance of thetail at 140 K thus suggests a few-meV (comparable to the 140K Boltzmann energy of 12 meV) defect state energy distribu-tion. Although in principle such defect state absorption shouldmanifest in photoluminescence spectra as well, no clear band-tailing was observed in low-temperature photoluminescencemeasurements . This is unsurprising, since above-gap ex-citation results in competing band-edge and defect state re-laxation pathways and emission Stokes shifts (on the orderof tens of meV in perovskite nanocrystals ) likely differfor defect transitions. For additional comparison, absorptionmeasurements were also performed on CsPbBr nanocubes(see Supplementary Material for absorption spectra and syn-thesis methods). Although a large anomalous band-gap shiftwas observed (approximately 40 meV from 6 to 140 K), noabsorption tail forms at low temperatures.Electron-phonon coupling that renormalizes the CsPbI bandgap should depend strongly on dimensionality innanocrystals. In particular, lowering dimensionality shouldreduce electron-phonon coupling by restricting certain vi-brational modes. To investigate the effect of lattice dimen-sionality on electron-phonon coupling, we repeat the sametemperature-dependent absorption measurements on CsPbI nanoplatelets. At room-temperature, a single nanoplatelet ab-sorption peak is observed that is blue-shifted relative to thenanocube band-gap due to strong quantum confinement inthe out-of-plane direction. At cryogenic temperatures, shownin Fig. 3, the absorption spectrum changes in two surprisingways. First, the nanoplatelet absorption peak continues nar-rowing below 140 K (with no absorption tail), in contrast tothe nanocube absorption peak width that remains constant atlow temperatures. Second, an additional lower-energy peakalso appears with decreasing temperature (see Fig. 3a) which,ffect of Dimensionality on the Optical Absorption Properties of CsPbI Perovskite Nanocrystals 4due to its center energy, we attribute to co-synthesized CsPbI nanocubes. Temperature-dependence surface plots of both thenanocube and nanoplatelet peaks (measured from the sameabsorption spectra) are shown in Fig. 3b to inform the relativechanges in peak optical density.Again fitting the nanoplatelet absorption peaks to Gaussianlineshapes, the fitted center energies are plotted in Fig. 3c. Thenearly-linear anomalous band-gap shift indicates weakenedelectron-phonon interactions and greater importance of band-gap renormalization due to unit cell expansion/contractionwith temperature. To quantify these changes, we perform alinear fit of the center energy temperature-dependence. Thefitted parameters are E = . A = . A is comparable to its corresponding nanocube value.Therefore, decreasing dimensionality greatly reduces vibra-tional band-gap renormalization without strongly affectingthat due to changes in unit cell size.The fitted Gaussian widths σ , plotted inset in Fig. 3c, re-veal another interesting aspect of the electronic properties ofperovskite nanoplatelets. While the nanocube absorption peakwidth is approximately constant at cryogenic temperatures, re-flecting its inhomogeneously broadened nature, the much nar-rower nanoplatelet absorption peak exhibits a monotonic de-crease in σ with decreasing temperature. This indicates thathomogeneous broadening in perovskite nanoplatelets con-tributes even down to cryogenic temperatures. However, aplateau in the linewidth decrease below 50 K, despite homo-geneous out-of-plane confinement, reveals an intrinsic ensem-ble absorption linewidth between 10 and 11 meV. At such en-ergy scales, inhomogeneous broadening due to variation in in-plane confinement of exciton center-of-mass motion, usuallyconsidered to be negligible , could become important. Moreadvanced spectroscopic techniques such as multi-dimensionalcoherent spectroscopy are needed to disentangle inhomo-geneous and homogeneous broadening mechanisms in per-ovskite nanoplatelets . IV. CONCLUSION
In summary, the absorption of CsPbI perovskite nanocrys-tals are measured at cryogenic temperatures. In addition to theanomalous band-gap shifts to higher energies with increasingtemperature, additional inflection points are observed at lowtemperatures that we attribute to band-gap renormalization by,contrary to a recent study , three vibrational modes in CsPbI nanocubes. Measurement of CsPbI nanoplatelets then re-veals greatly reduced vibrational band-gap renormalization,which suggests that lowered nanocrystal dimensionality leadsto weakened influence of lattice vibrations on electronic dy-namics. Lastly, absorption tails are found to form in CsPbI nanocubes at low temperatures, which we attribute to defectstates surrounding the valence band-edge. While perovskitenanocrystals have been found to be exceptionally defect-tolerant , our finding suggests that shallow defects may be-gin to influence the optical properties of iodide nanocubes atcryogenic temperatures. This work motivates further study ofelectron-phonon coupling in perovskite nanocrystals to mini- (a) ) . u . a ( n o i t p r o s b A (c) NPs(b) NPsNCs
FIG. 3. (a) Absorption spectra of CsPbI nanoplatelets at three rep-resentative temperatures 6, 80, and 140 K. In addition to the mainnanoplatelet (NP) absorption peak, a weak nanocube (NC) absorp-tion peak at lower energy appears at low temperatures. Inset showscomparison between 6 K and room-temperature absorption spectra.(b) Optical density surface plots of the NC and NP absorption peaksin (a) as a function of temperature. (c) NP center energies obtainedfrom the absorption peak first moment as a function of temperature,which reflects the material band-gap. A linear fit is plotted as thesolid blue curve. The fitted Gaussian widths are plotted inset, whichmonotonically decrease with decreasing temperature. ffect of Dimensionality on the Optical Absorption Properties of CsPbI Perovskite Nanocrystals 5mize their deleterious effects.
ACKNOWLEDGMENTS
This work was supported by the Department of Energygrant number de-sc0015782 and by the Sao Paulo ResearchFoundation, under the grant number 2013/16911-2. D.B.A.and G.N. acknowledge support by fellowships from theBrazilian National Council for Scientific and Technologi-cal Development (CNPq). Research was also supported byLNNano/CNPEM/MCTIC, where the TEM measurementswere performed. M. K. Choi, J. Yang, T. Hyeon, and D.-H. Kim, “Flexible quantum dotlight-emitting diodes for next-generation displays,” npj Flexible Electronics , 10 (2018). M. Yuan, M. Liu, and E. H. Sargent, “Colloidal quantum dot solids forsolution-processed solar cells,” Nature Energy , 16016 EP – (2016), reviewArticle. L. Protesescu, S. Yakunin, M. I. Bodnarchuk, F. Krieg, R. Caputo, C. H.Hendon, R. X. Yang, A. Walsh, and M. V. Kovalenko, “Nanocrystals ofcesium lead halide perovskites (cspbx , x = cl, br, and i): Novel optoelec-tronic materials showing bright emission with wide color gamut,” NanoLetters , 3692–3696 (2015). Y. Bekenstein, B. A. Koscher, S. W. Eaton, P. Yang, and A. P. Alivisatos,“Highly luminescent colloidal nanoplates of perovskite cesium lead halideand their oriented assemblies,” Journal of the American Chemical Society , 16008–16011 (2015). Y. Tong, E. Bladt, M. F. Aygüler, A. Manzi, K. Z. Milowska, V. A. Hinter-mayr, P. Docampo, S. Bals, A. S. Urban, L. Polavarapu, and J. Feldmann,“Highly luminescent cesium lead halide perovskite nanocrystals with tun-able composition and thickness by ultrasonication,” Angewandte ChemieInternational Edition , 13887–13892 (2016). M. J. Jurow, T. Morgenstern, C. Eisler, J. Kang, E. Penzo, M. Do, M. En-gelmayer, W. T. Osowiecki, Y. Bekenstein, C. Tassone, L.-W. Wang, A. P.Alivisatos, W. Brütting, and Y. Liu, “Manipulating the transition dipolemoment of cspbbr3 perovskite nanocrystals for superior optical properties,”Nano Letters , 2489–2496 (2019). Q. Wang, X.-D. Liu, Y.-H. Qiu, K. Chen, L. Zhou, and Q.-Q. Wang, “Quan-tum confinement effect and exciton binding energy of layered perovskitenanoplatelets,” AIP Advances , 025108 (2018). V. A. Hintermayr, A. F. Richter, F. Ehrat, M. Döblinger, W. Vanderlin-den, J. A. Sichert, Y. Tong, L. Polavarapu, J. Feldmann, and A. S. Urban,“Tuning the optical properties of perovskite nanoplatelets through composi-tion and thickness by ligand-assisted exfoliation,” Advanced Materials ,9478–9485 (2016). M. C. Weidman, A. J. Goodman, and W. A. Tisdale, “Colloidal halideperovskite nanoplatelets: An exciting new class of semiconductor nanoma-terials,” Chemistry of Materials , 5019–5030 (2017). S. Peng, S. Wang, D. Zhao, X. Li, C. Liang, J. Xia, T. Zhang, G. Xing,and Z. Tang, “Pure bromide-based perovskite nanoplatelets for blue light-emitting diodes,” Small Methods , 1900196 (2019). M. Wei, F. P. G. de Arquer, G. Walters, Z. Yang, L. N. Quan, Y. Kim,R. Sabatini, R. Quintero-Bermudez, L. Gao, J. Z. Fan, F. Fan, A. Gold-Parker, M. F. Toney, and E. H. Sargent, “Ultrafast narrowband excitonrouting within layered perovskite nanoplatelets enables low-loss lumines-cent solar concentrators,” Nature Energy , 197–205 (2019). P. Cottingham and R. L. Brutchey, “On the crystal structure of colloidallyprepared cspbbr3 quantum dots,” Chemical Communications , 5246–5249 (2016). F. Bertolotti, L. Protesescu, M. V. Kovalenko, S. Yakunin, A. Cervel-lino, S. J. L. Billinge, M. W. Terban, J. S. Pedersen, N. Masciocchi, andA. Guagliardi, “Coherent nanotwins and dynamic disorder in cesium leadhalide perovskite nanocrystals,” ACS Nano , 3819–3831 (2017). R. J. Sutton, M. R. Filip, A. A. Haghighirad, N. Sakai, B. Wenger,F. Giustino, and H. J. Snaith, “Cubic or orthorhombic?: Revealing the crys- tal structure of metastable black-phase cspbi3 by theory and experiment,”ACS Energy Letters , 1787–1794 (2018). L. Protesescu, S. Yakunin, S. Kumar, J. Bär, F. Bertolotti, N. Mascioc-chi, A. Guagliardi, M. Grotevent, I. Shorubalko, M. I. Bodnarchuk, C.-J.Shih, and M. V. Kovalenko, “Dismantling the "red wall" of colloidal per-ovskites: Highly luminescent formamidinium and formamidinium-cesiumlead iodide nanocrystals,” ACS Nano , 3119–3134 (2017). L. G. Bonato, R. F. Moral, G. Nagamine, A. A. de Oliveira, J. C. Germino,D. S. da Silva, F. Galembeck, L. A. Padilha, and A. F. Nogueira, “Soon tobe published,”. X. Sheng, G. Chen, C. Wang, W. Wang, J. Hui, Q. Zhang, K. Yu, W. Wei,M. Yi, M. Zhang, Y. Deng, P. Wang, X. Xu, Z. Dai, J. Bao, andX. Wang, “Polarized optoelectronics of cspbx3 (x = cl, br, i) perovskitenanoplates with tunable size and thickness,” Advanced Functional Materi-als , 1800283 (2018). K. Momma and F. Izumi, “
VESTA3 for three-dimensional visualization ofcrystal, volumetric and morphology data,” Journal of Applied Crystallogra-phy , 1272–1276 (2011). A. Liu, D. B. Almeida, W. K. Bae, L. A. Padilha, and S. T. Cundiff, “Non-markovian exciton-phonon interactions in core-shell colloidal quantum dotsat femtosecond timescales,” Physical Review Letters , 057403 (2019). A. Göbel, T. Ruf, M. Cardona, C. T. Lin, J. Wrzesinski, M. Steube,K. Reimann, J.-C. Merle, and M. Joucla, “Effects of the isotopic composi-tion on the fundamental gap of cucl,” Physical Review B , 15183–15190(1998). I.-H. Choi and P. Y. Yu, “Suppression of the anomalous blue shift in theband gap temperature dependence of agcugas alloys,” Physical Review B , 235210 (2001). C. Yu, Z. Chen, J. J. Wang, W. Pfenninger, N. Vockic, J. T. Kenney, andK. Shum, “Temperature dependence of the band gap of perovskite semicon-ductor compound cssni ,” Journal of Applied Physics , 063526 (2011). R. Saran, A. Heuer-Jungemann, A. G. Kanaras, and R. J. Curry, “Gi-ant bandgap renormalization and exciton-phonon scattering in perovskitenanocrystals,” Advanced Optical Materials , 1700231 (2017). M. Cardona, “Electron-phonon interaction in tetrahedral semiconductors,”Solid State Communications , 3–18 (2005). N. Moses Badlyan, A. Biermann, T. Aubert, Z. Hens, and J. Maultzsch,“Thermal expansion of colloidal cdse/cds core/shell quantum dots,” Physi-cal Review B , 195425 (2019). N. Garro, A. Cantarero, M. Cardona, A. Göbel, T. Ruf, and K. Eberl, “De-pendence of the lattice parameters and the energy gap of zinc-blende-typesemiconductors on isotopic masses,” Physical Review B , 4732–4740(1996). I.-H. Choi, S.-H. Eom, and P. Y. Yu, “Soft phonon mode and the anomaloustemperature dependence of band gap in aggas2,” physica status solidi (b) , 99–104 (1999). M. A. Pérez-Osorio, R. L. Milot, M. R. Filip, J. B. Patel, L. M. Herz, M. B.Johnston, and F. Giustino, “Vibrational properties of the organic-inorganichalide perovskite ch nh pbi from theory and experiment: Factor groupanalysis, first-principles calculations, and low-temperature infrared spec-tra,” The Journal of Physical Chemistry C , 25703–25718 (2015). C. Yin, L. Chen, N. Song, Y. Lv, F. Hu, C. Sun, W. W. Yu, C. Zhang,X. Wang, Y. Zhang, and M. Xiao, “Bright-exciton fine-structure splittingsin single perovskite nanocrystals,” Physical Review Letters , 026401(2017). G. Raino, G. Nedelcu, L. Protesescu, M. I. Bodnarchuk, M. V. Ko-valenko, R. F. Mahrt, and T. Stoferle, “Single cesium lead halide perovskitenanocrystals at low temperature: Fast single-photon emission, reducedblinking, and exciton fine structure,” ACS Nano , 2485–2490 (2016). H. Qiao, K. A. Abel, F. C. J. M. van Veggel, and J. F. Young, “Excitonthermalization and state broadening contributions to the photoluminescenceof colloidal pbse quantum dot films from 295 to 4.5 k,” Physical Review B , 165435 (2010). I. Studenyak, M. Kranjec, and M. Kurik, “Urbach rule in solid statephysics,” International Journal of Optics and Applications , 76–83 (2014). P. Guyot-Sionnest, E. Lhuillier, and H. Liu, “A mirage study of cdse col-loidal quantum dot films, urbach tail, and surface states,” The Journal ofChemical Physics , 154704 (2012). J. Kang and L.-W. Wang, “High defect tolerance in lead halide perovskitecspbbr3,” The Journal of Physical Chemistry Letters , 489–493 (2017). ffect of Dimensionality on the Optical Absorption Properties of CsPbI Perovskite Nanocrystals 6 B. I. Halperin and M. Lax, “Impurity-band tails in the high-density limit. i.minimum counting methods,” Physical Review , 722–740 (1966). R. Yan, W. Zhang, W. Wu, X. Dong, Q. Wang, and J. Fan, “Optical spec-troscopy reveals transition of cuins /zns to cu x zn − x ins /zns:cu alloyedquantum dots with resultant double-defect luminescence,” APL Materials , 126101 (2016). J. Jean, T. S. Mahony, D. Bozyigit, M. Sponseller, J. Holovský, M. G.Bawendi, and V. Bulovic, “Radiative efficiency limit with band tailing ex-ceeds 30% for quantum dot solar cells,” ACS Energy Letters , 2616–2624(2017). M. Nasilowski, B. Mahler, E. Lhuillier, S. Ithurria, and B. Dubertret, “Two-dimensional colloidal nanocrystals,” Chemical Reviews , 10934–10982 (2016). S. T. Cundiff and S. Mukamel, “Optical multidimensional coherent spec-troscopy,” Physics Today , 44–49 (2013). A. Liu, D. B. Almeida, W.-K. Bae, L. A. Padilha, and S. T. Cundiff, “Si-multaneous existence of confined and delocalized vibrational modes in col-loidal quantum dots,” The Journal of Physical Chemistry Letters , 6144–6150 (2019). H. Huang, M. I. Bodnarchuk, S. V. Kershaw, M. V. Kovalenko, and A. L.Rogach, “Lead halide perovskite nanocrystals in the research spotlight: Sta-bility and defect tolerance,” ACS Energy Letters2