Engineering Carrier Dynamics in Halide Perovskites by Dynamical Lattice Distortion

Abstract The electronic structure of halide perovskites is central to their carrier dynamics, enabling the excellent optoelectronic performance. However, the experimentally resolved transient absorption spectra exhibit large discrepancies from the commonly computed electronic structure by density functional theory. Using pseudocubic CsPbI3 as a prototype example, here, it is unveiled with both ab initio molecular dynamics simulations and transmission electron microscopy that there exists pronounced dynamical lattice distortion in the form of disordered instantaneous octahedral tilting. Rigorous first‐principles calculations reveal that the lattice distortion substantially alters the electronic band structure through renormalizing the band dispersions and the interband transition energies. Most notably, the electron and hole effective masses increase by 65% and 88%, respectively; the transition energy between the two highest valence bands decreases by about one half, agreeing remarkably well with supercontinuum transient‐absorption measurements. This study further demonstrates how the resulting electronic structure modulates various aspects of the carrier dynamics such as carrier transport, hot‐carrier relaxation, Auger recombination, and carrier multiplication in halide perovskites. The insights provide a pathway to engineer carrier transport and relaxation via lattice distortion, enabling the promise to achieve ultrahigh‐efficiency photovoltaic devices.


Introduction
Halide perovskites are exceptional candidates for photovoltaic and light-emitting devices, with a champion solar conversion efficiency approaching 26%. [1]The remarkable optoelectronic performance is intimately related to their microscopic electronic structure, which determines basic properties such as bandgap and effective masses, and derived properties such as absorption, luminescence, carrier mobilities, and carrier recombination rates. [2,3]xperimental techniques such as angleresolved photoelectron spectroscopy (ARPES), [4,5] allow to directly probe the electronic band structure of the valence bands.However, due to complications from sample details and experimental conditions, it could be very challenging to experimentally map the entire band structure.For example, ARPES experiments have led to contradictory results about the giant Rashba splitting in the valence band (VB) of lead-halide perovskites. [6,7]y contrast, first-principles calculations can overcome this limitation and have been routinely employed to compute the electronic band structures of various materials.In particular, with the developments of Y. Han  accurate hybrid exchange-correlation functional and many-body approaches such as GW, the electronic band structure can be theoretically determined precisely. [8][11][12][13] Reliable approaches based on the Heyd-Scuseria-Ernzerhof (HSE) [14] hybrid functional [15][16][17] or GW [18,19] in conjunction with spin-orbit coupling (SOC) have been established.However, an accurate assessment of the electronic structure is only possible if correct atomic structures for halide perovskites have been employed as input.
Using both theory and experiment for a prototypical halide perovskite CsPbI 3 , we show that the atomic structure of pseudocubic CsPbI 3 exhibits pronounced dynamical lattice distortion.The pseudocubic phase of CsPbI 3 at room temperature is only cubic on average; the instantaneous atomic structure contains disordered octahedral tilting.With more realistic atomic structures, we demonstrate that the electronic band structure is substantially altered.The effective masses and interband transition energies reveal giant renormalization, which ultimately affect the carrier dynamics (e.g., transport, relaxation, recombination, and multiplication) in halide perovskites.Our study suggests that lattice distortion can be employed to modulate the carrier dynamics and ultimately the optoelectronic performance.

Discrepancy Between Experiment and Theory
Figure 1a presents the experimentally measured absorption spectrum of black-phase CsPbI 3 nanocrystals (see Experimental Section for synthetic details).The band-edge absorption peak is at ∼670 nm (or 1.85 eV), which is slightly blue-shifted compared to that of bulk black-phase CsPbI 3 because of quantum confinement as a result of its finite nanocrystal size (∼9 nm).In addition to the band-edge feature, there exist two extra absorption "bumps" at 440 and 365 nm, which are associated with higherenergy electronic bands.
The absorption features can be better visualized and assigned using broadband transient absorption (TA) spectra covering ultraviolet to near-infrared spectral regions.The TA spectra in Figure 1b are acquired with 660 nm pump, which directly address the band-edge transition.Note that our samples are colloidal dispersions of monodisperse perovskite nanocrystals with size <10 nm, and therefore there exists negligible optical scattering, as proved by the TA spectrum plotted at a negative pump-probe delay.In addition to the band-edge bleach feature (labeled as "B1"), the two bleach features associated with the absorption "bumps" above are also present ("B2" and "B3").Because TA bleach arises from state-filling effects of photogenerated carriers, simultaneous observation of B2 and B3 bleaches upon exciting into B1 indicates that they share a common state from the conduction-band minimum (CBM) or valence-band maximum (VBM) with B1.On the basis of the spin selection rules, a previous study has assigned B3 to a transition from the VBM to the second-lowest CB (CBM+1), which is split from the CBM by SOC. [20]Specifically, when the CsPbI 3 nanocrystals are excited by a  + (left-handed circular polarization) photon resonant with B1, B1 has a maximum bleach under a  + probe photon but a negligible bleach under a  − (right-handed circular polarization) probe photon, and B3 has a maximum bleach under a  − probe photon.This is consistent with the fact that B3 is dominated by the transition from the VBM (m h = ±1/2) to the heavy-electron level in the CBM+1 (m j = ±3/2) that is separated from CBM (m j = ±1/2) by the SOC splitting.This assignment is further substantiated by the gap of 1.53 eV between B1 and B3 matching the well-accepted SOC value of ∼1.5 eV for lead-halide perovskites. [3,21]By contrast, B2 probed in the same experiment shows no spin selectivity.It cannot be associated with a transition from the VBM to a level in the CB, because, at the experimentally observed energy separation of ∼1 eV, there is no level in the CB.This leaves B2 to be likely associated with a transition from the second-highest VB (VBM+1) to CBM.
The observation of three bleach features is not specific to the nanocrystal size or A-site (ABX 3 ) cation composition.All these features are observed in different-size CsPbI 3 nanocrystals as well as in FAPbI 3 (FA: formamidinium) nanocrystals (see Figure S1, Supporting Information).Crucially, the B1-B2 and B1-B3 separations of ∼1.5 and ∼1.0 eV, respectively, are universal to these samples.The studied CsPbI 3 perovskite nanocrystals are in the size range of 5-9 nm, which is in the intermediate-to-weak confinement regime. [22]Although the band-edge transition (B1) indeed blueshifts from ca. 665 to 635 nm with decreasing nanocrystal sizes, the B1-B2 and B1-B3 energy separations are less sensitive to the nanocrystal sizes, likely because the CBM and CBM+1 (or VBM and VBM+1) are shifted by the confinement effect in a similar way.Previous reports on MAPbI 3 (MA: methylammonium) films revealed a similar B1-B2 separation of ∼1 eV, [23,24] although the B3 feature was missing due to limited spectral width of the probe beam in those studies.Above all, these results suggest that B1-B2 and B1-B3 separations are intrinsic to lead-iodide perovskites, and that quantum confinement and A-site cation have weak impacts on these energies.In principle, accommodating different sizes of A-site cations should lead to differences in these energy separations, but these differences might be too small to be fully captured by our TA measurements.
The extracted transition energies enable a careful benchmark of the electronic band structure from first principles.Figure 1c shows the band structure of cubic CsPbI 3 by using accurate HSE+SOC scheme with a fraction of the Fock exchange that allows to reproduce the the experimental bandgap (≈1.73 eV) for bulk CsPbI 3 thin films at room temperature [25,26] (see Experimental Section for details).Although the bandgap is correct, the intra-VB (ΔE v ) and intra-CB (ΔE c ) transition energies are substantially different from the experimental values.With our firstprinciples calculations, we obtain 1.72 eV for ΔE c and 2.20 eV for ΔE v .

Evidence for Dynamical Lattice Distortion
The large discrepancy between theory and experiment in the intra-VB and intra-CB transition energies sets an interesting puzzle: what is missing in the first-principles calculations?The HSE+SOC scheme has, in principle, proven great accuracy and reliability in computing the electronic structure of halide perovskites. [27]However, we notice that dynamical lattice distortion has been experimentally reported for halide perovskites at room temperature. [28,29]Although often being ignored in previous computations, this lattice distortion might have important consequences for the electronic band structure.
We perform structural characterizations to identify the lattice distortion in the CsPbI 3 nanocrystals at around room temperature.Figure 2a shows a high-resolution transmission electron microscope (TEM) image of a typical CsPbI 3 nanocrystal.The nanocrystal has a cubic shape with an edge-length of 10 nm, and the overall lattice is pseudocubic-like.Scrutiny of the image, however, reveals substantial lattice distortion, arising from tilting of Pb-I octahedra (see the enlarged image in Figure 2a).In principle, Fast Fourier Transform (FFT) analysis of the highresolution TEM images may yield quantitative information on the lattice distortion; however, it is infeasible here due to sample instability under electron beams.In our recent study, [22] we applied temperature-dependent X-ray diffraction (XRD) to quantify this lattice distortion.The orthorhombic lattice constants b and a were found to increase and decrease with decreasing temperature, respectively, indicative of an enhanced lattice distortion with decreasing temperature.In spite of this lattice distortion, however, the overall crystal framework formed by corner-sharing PbI 6 octahedra is still preserved, and the lattice constant changes can be treated as a primarily orthorhombic strain applied to the cubic phase.
We note that the TEM image provides averaged results of the lattice; many instantaneous configurations are averaged during the acquisition time of a few seconds.Thus, these results serve only as a qualitative evaluation of the dynamical lattice distortion.Directly tracking single-particle lattice distortion on ultrafast timescales is beyond the current experimental capability.In order to extract more realistic atomic structure of CsPbI 3 in real time, we therefore perform ab initio molecular dynamics (AIMD) simulations of CsPbI 3 at room temperature.The initial configuration is a 3 ×3 ×3 supercell of the ideal cubic perovskite structure of CsPbI 3 .Using this configuration as the starting point, we can observe and analyze the structural evolution of the cubic phase during the simulation, as well as their impact on the electronic band structure.Figure 2b shows the evolution of the energy gaps at different high-symmetry k-points (Γ and R) as a function of the AIMD simulation time.The energy gap at Γ quickly decreases and the gap at R (the actual bandgap) gradually increases within the first 1000 fs.This clearly demonstrates that lattice vibrations and distortions at room temperature substantially impact the electronic structure and indicates the instability or lack of relaxation of the initial cubic structure.However, once thermodynamic equilibrium has reached (after ∼1000 fs), the structures is still changing in every step, but the lattice is always on a similar level of distortion from the ideal cubic perovskite (Figure S3, Supporting Information), which leads to similar electronic-structure renormalization.Notably, the observed behavior is almost independent of the initial structure selected for the simulation.We simulate the X-ray diffraction (XRD) patterns of the AIMD configurations as depicted in Figure 2c, which can be compared with the experimental result for CsPbI 3 measured at 303 K (upper panel in Figure 2c).The XRD pattern begins with the one for ideal cubic CsPbI 3 , but very quickly additional diffraction peaks start to develop due to lattice distortions and remain similar after ∼1000 fs.The new sharp XRD peaks are partly due to the finite size of the supercell.However, these new features in the XRD pattern are indeed signature for dynamical lattice distortion, which can be clearly observed in the experimental XRD spectrum as well (most notably, the ones within 20°a nd 30°).The good agreement between theory and experiment demonstrates again the presence of symmetry breaking induced by dynamical octahedral tilting in pseudocubic CsPbI 3 .

Impact of Dynamical Lattice Distortion on Band Structure
To quantitatively study the impact of dynamical lattice distortion on the electronic structure, we select four representative con-figurations from the AIMD trajectory.The four configurations are randomly sampled from four time intervals (1000−2000 fs, 2000−3000 fs, 3000−4000 fs, and 4000−5000 fs), with a criterion that their bandgaps at R are close to the averaged value after equilibration. Figure 3a-d shows their atomic structures; these structures retain the overall cubic shape, but consistently exhibit severe distortions characterized by Pb-I octahedral tilting.Then the key question is how would these structural distortions modulate the electronic structure.
The local octahedral tilting induces lattice distortions and also breaks the cubic symmetry of the lattice.Hence, to better present and analyze the electronic structure of the pseudocubic CsPbI 3 , we compute the band structures of the four representative configurations using accurate HSE+SOC scheme and unfold the band structure to the cubic Brillouin zone of the unit cell. [30]The corresponding unfolded band structures are shown in Figure 3e-h.Three important insights can be extracted from these results.
First, there exists substantial band renormalization for the distorted configurations.As compared to the band structure of the ideal cubic CsPbI 3 (yellow lines), the renormalized band structure shows drastically reduced intra-VB and intra-CB transition energies.In particular, ΔE v decreases by about one half.The reduced transition energies now actually agree very well with the experimentally determined values, demonstrating that the local structural distortions are responsible for the discrepancy between theory and experiment observed above.
Second, for both the lowest CB and the highest VB the dispersions are severely altered, leading to heavier effective masses for both electrons and holes.The electron effective mass is enhanced by 65% (0.119 m 0 vs 0.196 m 0 ), and the hole effective mass increases by 88% (0.119 m 0 vs 0.224 m 0 ).We note that to reproduce the experimental bandgap of 1.73 eV, we used an HSE mixing of 0.53 for the pseudocubic structure at 300 K, which is smaller than the one used for the ideal cubic structure (0.87).However, as shown in Figure S2 and Table S1 (Supporting Information), if we use a consistent HSE mixing of 0.53 for the ideal cubic structure, the electron and hole effective masses are even smaller (0.102 m 0 for electrons and 0.100 m 0 for holes).The corresponding intra-VB and intra-CB transition energies become 1.78 and 1.64 eV, respectively, which are still much larger than the experimentally measured ΔE v (1.01 eV) and ΔE c (1.53 eV).This comparison demonstrates that the giant band renormalization is not caused by the large HSE mixing used for the ideal cubic structure.
Third, despite the fact that the four representative configurations have different local atomic structures, their electronic band structures are extremely similar.This implies that there is a weak dependence of the band structure on the configuration cho-sen in the AIMD simulations, once dynamical equilibrium has reached.The consistency between the unfolded band structures also means that our computed band structures well represent the electronic structure of distorted CsPbI 3 at room temperature.We note that even for configurations with bandgaps slightly different from the averaged value after equilibration, their band structures are also very similar to the ones in Figure 3e-h.
Dynamical lattice distortion also affects the localization of wavefunctions.As shown in Figure S4 (Supporting Information), the initial wavefunctions in the cubic CsPbI 3 display regular and uniform patterns at the VBM and CBM.As the simulation proceeds, partial localization of wavefunctions due to lattice distortion becomes apparent.However, the induced quantum wells from lattice distortion are shallow, resulting in relatively weak wavefunction localization.
We further investigate the correlation between structural distortion and band renormalization in order to understand the microscopic mechanisms.As mentioned above, the structural distortions are primarily characterized by local dynamical tilting of the Pb-I octahedra.The octahedral tilting is well-known for perovskites, and is also a key ingredient of the structural transitions from the cubic phase to the tetragonal or orthorhombic phases. [31]As shown in Figure 4a, the Pb-I octahedra only need to collectively rotate around the c-axis in order to transition from the cubic phase (Pm3m) to the tetragonal (P4/mbm) phase (marked 1 tilt, a 0 a 0 c − , following the Glazer notation [32] ); there is a rotation angle  1 .For the transition between the cubic and orthorhom- bic (Pnma) phases, the Pb-I octahedra rotate around three axes, which is labeled 3 tilts, a − a − b + .There are two distinct rotation angles:  1 (in-plane) and  2 (out-of-plane).It is difficult to examine the tilting of a single octahedron and to take into account the dynamical disordering effect.Hence, we instead analyze the impact of collective octahedral tilting on the electronic structure during the cubic-to-tetragonal and cubic-to-orthorhombic phase transitions; the fundamental mechanisms are the same.For this analysis, we constrain the overall cubic shape, but allow Pb-I octahedra to tilt in order to transition from the cubic phase to the "tetragonal" or "orthorhombic" phases.The rotation angle for the cubic-to-tetragonal transition is 13.32°( 1 ), while the two rotation angles for the cubic-to-orthorhombic transition are 12.00°(  1 ) and 8.36°( 2 ).
We analyze the variation of the energies as a function of a generalized configuration coordinate (Q) that characterizes the structural transformation from the cubic to tetragonal or orthorhombic phases.Q is defined as the atomic mass (m i ) weighted Cartesian coordinate (R i ) with respect to the reference configuration- [27] As depicted in Figure 4b, with the increase in the structural distortion (i.e., the octahedral tilting angle) during the cubic-to-tetragonal transition the VBM level shows a pronounced decrease and the CBM level slightly increases.The CBM+1 energy remains almost unchanged, while the VBM+1 energy gradually deceases with increasing distortion; but the decrease in the VBM+1 energy is less pronounced than that in the VBM.Similar behavior is also found for the cubic-to-orthorhombic phase transition (Figure 4c).
Figure 4d,e plot variation of the transition energies (E g , ΔE c , and ΔE v ) in CsPbI 3 during the cubic-to-tetragonal and cubicto-orthorhombic phase transitions.Obviously, E g gradually increases, similar to the trend as observed in the first 1000 steps of the AIMD simulations.Both ΔE c and ΔE v decrease rapidly with increasing the octahedral tilting.Our detailed analysis of the atomic structures in Figure 3a-d shows that the local rotation angle of Pb-I octahedra can reach up to 24.35°, which is 2-3 times larger than the ones associated with the cubic-to-tetragonal and cubic-to-orthorhombic transitions.Also, in pseudocubic CsPbI 3 at room temperature there exists octahedral tilting in three principal axes.The large tilting of Pb-I octahedra leads to substantial reduction of ΔE c and ΔE v , which explains the giant band renormalization for the pseudocubic CsPbI 3 at room temperature observed in our first-principles calculations (Figure 3e-h) and the large discrepancy between theory and experiment (Figure 1a,b).
The variations of the VBM and CBM energies and transition energies can be understood by analyzing the orbital diagram of CsPbI 3 in three different phases.As schematically shown in Figure 4f, the VBM of CsPbI 3 perovskites is an antibonding state between Pb 6s and I 5p orbitals, and the VBM+1 is a bonding state between the Pb 6p and I 5p orbitals. [33]Both the CBM and CBM+1 are antibonding states of Pb 6p and I 5p orbitals, [34] but they are split due to strong SOC and local crystal field.As the octahedron tilts, the local symmetry is reduced, leading to reduced s-p coupling (between VBM and lower-lying Pb 6s states) and enhanced p-p coupling (between CBM/CBM+1 and VBM/VBM+1). [35]The enhanced p-p coupling increases the CBM and CBM+1 energies, while decreases the VBM and VBM+1 energies.The reduced s-p coupling does not affect the CBM and VBM+1 energies, but further lowers the VBM energy.The octahedral tilting also increases the Pb-I bond length, which, as a secondary effect, decreases the CBM and VBM energies, and slightly increases the VBM+1 energy.Consequently, the energy difference between the VBM and VBM+1 (ΔE v ) decreases.As discussed above, the CBM+1 energy should, in principle, increase due to enhanced p-p coupling.Nevertheless, while the octahedron rotates, the p x , p y , and p z orbitals start to mix, which results in an effective lowering of the SOC splitting.Hence, the CBM+1 energy slightly decreases, and the energy difference between CBM and CBM+1 (ΔE c ) is also reduced.

Implications for Carrier Dynamics
The giant band renormalization has direct relevance to the carrier dynamics in lead-iodide perovskites.The most straightforward consequence is that the carrier transport would be much slower.As we discussed before, the giant band renormalization leads to increases in both the electron and hole effective masses by 65% and 88%, respectively.The band dispersions as characterized by the carrier effective masses directly influence the carrier mobilities.Within the deformation potential theory, the carrier mobility () relates to the effective mass (m * ) by [36]  = 2 where e is elementary charge, ℏ the Planck constant, c ii the elastic constant, k B the Boltzmann constant, T the absolute temperature, and E 1 a deformation-potential constant.Based on Equation (1), one can easily estimate that a 65% (88%) increase in the electron (hole) effective mass would decrease the electron (hole) mobility by 72% (89%), which is a very pronounced effect and would substantially impact the device performance.Actually, previous first-principles calculations [37] have found that the explicitly computed carrier mobilities for cubic CsPbI 3 are higher than the experimental values [38,39] by around one order of magnitude.
While there are many factors that could be responsible for the large difference, the giant band renormalization might be a relevant mechanism.The suppressed carrier transport by dynamical lattice distortion might also provide an intuitive explanation to the unexpectedly low carrier mobilities in halide perovskites observed in experiments. [40]With the identification of a possible cause of low carrier mobilities, effective strategies might be developed to improve carrier transport in halide perovskites via lattice and band-structure engineering.Further, the band renormalization pushes the VBM+1-to-CBM transition from the ultraviolet to the visible spectral region (see Figure 3e-h), and hence increases the material absorption in the visible.Although the absorption coefficients of these halide perovskites are often sufficiently large, this enhancement might still be important when the light-absorption layer has a limited thickness.
As to the dynamical aspect, the strong reduction of the intra-VB transition energy (ΔE v ) might significantly alter the hotcarrier relaxation dynamics.In the scenario of cubic CsPbI 3 , bluelight excitation creates highly-energetic carriers at high-k states in the lowest CB and highest VB.These hot carriers can rapidly relax by emitting phonons.Now with the distorted CsPbI 3 , bluelight excitation generates hot holes in the dispersion curve of VBM+1 (see Figure 5a).If the energy-momentum mismatch between VBM+1 and the highest VB is not easily accommodated by phonons, there might be long-sustained hot holes in VBM+1, which could be harvested for efficient hot-carrier devices.We design pump wavelength-dependent TA experiments to verify this hypothesis.As plotted in Figure 5b, we adopt four different pump wavelengths, among that 660 nm is slightly above the VBM-to-CBM transition; 450 and 440 nm are in resonance with the VBM+1-to-CBM transition, whereas 405 nm is off resonance with the VBM+1-to-CBM transition and mostly contributed by high-k VB-to-CB transitions.
The broadband TA spectra are presented in Figure S5 (Supporting Information).All the spectra are controlled to have approximately identical signal sizes in order to ensure a similar level of carrier occupancies with different pump wavelengths.The major distinction between 660 nm and other shorter wavelength excitations lies in the first few ps.Short-wavelength excitations result in state-filling bleaches and photoinduced absorption features on the blue and red sides, respectively, of the lowest bleach peak (Figure S5, Supporting Information), a manifestation of hot-carrier population. [41,42]These features disappear within ∼2 ps, indicating that the majority of the hot carriers relax to CBM and VBM within this time.This ultrafast hot-carrier cooling is a typical observation that has been extensively studied in previous studies.Therefore, we will not elaborate on this here.Instead, here we focus on a minor, slow part of the hotcarrier relaxation.In principle, we should be able to directly track hot-carrier dynamics by probing the B2 feature at ca. 420 nm.However, strong absorptions of the sample near this region result in strong attenuation of the probe light and hence very poor (or even undetectable) transient absorption signals near B2.Further, as shown in Figure 1b, the B2 feature is present even if we pump resonantly into the B1 feature, as they share the common CBM level, that is, the B2 feature contains significant contribution from band-edge carriers.These two issues, in combination with the very low portion of long-lived hot carriers, prohibit direct measurement of hot-hole relaxation by following the B2 kinetics.Instead, we choose to monitor the carrier dynamics probed near the band edge where the signal/noise ratio of the transient absorption signals is optimal.Specifically, if we normalize the TA spectra from ∼4 to 60 ps to their bleach maxima, we find that the spectra still differ for different pump wavelengths.The 660 nm excited spectra have essentially no spectral evolution on this timescale (Figure 5c), as expected for the absence of hot-carrier relaxation under this pump wavelength.By contrast, the 440 nm excited spectra are well normalized to the red side but show progressive spectral narrowing on the blue side (Figure 5d), which can be better seen in the enlarged view to the right.This is a fairly reproducible phenomenon that can be observed with 450 nm excitation as well (Figure S6, Supporting Information).[45] Most importantly, under 405 nm excitation, the spectral narrowing on the blue side is less prominent than the 440 nm case (Figure 5e).In principle, under a similar level of carrier occupancies, one should expect a slower hot-carrier relaxation for 405 nm excitation, as in this case the carriers simply have more energies to dissipate.
The counterintuitive behavior, however, can be well rationalized by invoking the VBM+1-to-CBM transition mentioned above.Because 440 and 450 nm are in resonance with this transition, they excite hot holes into VBM+1 and these hot-holes display slow relaxation likely due to the energy-momentum mismatch between VBM+1 and the highest VB speculated above (Figure 5a), that is, the energy and momentum released in this transition cannot be simultaneously accommodated by one phonon (most likely the longitudinal optical phonon).Because hot carriers relax in a cascade-like fashion, this "bottleneck" also slows down the ensuing relaxation events, eventually being manifested as slowly-decaying blue-side bleaches in Figure 5d and Figure S6 (Supporting Information).By contrast, 405 nm excitation is off resonance with the VBM+1-to-CBM transition and becomes dominated by high-k VB-to-CB transitions; the resulting hot carriers can rapidly relax by emitting phonons.
While we cannot exclude the existence of such a hot-hole relaxation "bottleneck" in an ideal cubic CsPbI 3 , the key is that lattice distortion has brought this effect into a solar-relevant regime (near 440 nm excitation).Otherwise, according to our calculation, in the ideal cubic structure this transition will be situated at >3.9 eV (Figure 1b), which is hard to access by solar photons.Although the contribution of the VBM+1-to-CBM transition to the overall optical absorbance in the range remains small, and therefore the slow hot-hole relaxation has a small amplitude, it is still a very promising observation as it opens a new avenue of engineering hot-carrier relaxation dynamics through lattice distortion and band renormalization in halide perovskites.It also offers an alternate explanation to the slow hot-carrier relaxation dynamics observed in these materials that have been generally explained using a polaronic screening mechanism. [46]esides hot-carrier cooling, the strong band renormalization might also have important consequences for three-carrier Auger recombination that involves excitation of a carrier into highenergy states. [21,47,48]The structure of high-energy bands dictates the transition momentum and density of the final states, and hence controls the rate of Auger recombination.In a similar way, the structure of high-energy bands could influence the "inverse" process of Auger recombination-carrier multiplication, [47,49,50] a process by which the photovoltaic efficiency can be strongly enhanced.Verification of these consequences is considerably more complex than hot-carrier cooling, but certainly warrants future efforts.Overall, many possibilities might be opened by the lattice distortion and strong band renormalization in halide perovskites.

Conclusion
To conclude, we have unveiled a giant electronic band renormalization in pseudocubic halide perovskite CsPbI 3 due to dynamical lattice distortion in the form of octahedral tilting, which has been commonly overlooked.The electronic band renormalization critically influences the carrier dynamics, which enables a new pathway to engineer carrier transport and relaxation via lattice distortion.This conclusion is expected to be applicable to both allinorganic and organic-inorganic hybrid perovskites (Figure S7, Supporting Information).We call for a reassessment of the previously computed electronic structures of halide perovskites, and properties that sensitively depend the electronic band structure such as effective masses, absorption, luminescence, carrier mobilities, and carrier recombination rates.

Experimental Section
First-Principles Calculations: All first-principles calculations in this work were performed using the projector augmented-wave (PAW) [51] pseu-dopotentials as implemented in the Vienna ab initio simulation package (VASP).The experimental lattice constant of 6.17 Å for cubic CsPbI 3 was adopted. [25]The HSE [14] hybrid functional including the SOC effect was employed for the first-principles calculations.A mixing parameter of 0.87 for the Fock exchange was used for the ideal cubic CsPbI 3 in order to reproduce the experimental bandgap, 1.73 eV. [25]The plane-wave energy cutoff was set to 500 eV and and a 6 ×6 ×6 Monkhorst-Pack [52] k-point grid was adopted for the cubic unit cell of CsPbI 3 .
A 3× 3×3 supercell of cubic CsPbI 3 containing 135 atoms was used for the AIMD simulations.For computational affordability, the AIMD simulations were carried out using the functional by Perdew, Burke, and Ernzerhof (PBE), [53] and SOC was not included.A canonical ensemble (NVT) with the Nosé-Hoover thermostat [54,55] at 300 K was employed.The time step for the AIMD simulations was 1 fs.The PBE+SOC scheme was utilized to calculate the partial charge density at the R point for selected configurations from the AIMD trajectory.A time step of 0.5 ps was employed to capture the dynamic changes in the wavefunction.This approach allowed to effectively visualize the spatial variations of the wavefunction throughout the simulation.A mixing parameter of 0.53 within the HSE+SOC scheme was used for the supercell band-structure calculations, which reproduces the experimental bandgap after unfolding.For unfolding the electronic band structures of disordered supercells, the scheme implemented in VASPKIT was employed. [56]o simulate the cubic-to-tetragonal and cubic-to-orthorhombic phase transitions, a √ 2 × √ 2×2 supercell was built from the cubic unit cell.This study adopted the same plane-wave energy cutoff and a 4 × 4 × 3 k-point grid for the HSE+SOC calculations.
Sample Synthesis and Characterizations: The black-phase CsPbI 3 nanocrystals were synthesized using a hot-injection method reported in previous studies. [48]For TEM characterizations, the nanocrystals in hexane were dropped onto ultrathin carbon TEM grids.The images were acquired on a JEOL JEM-2100 at 200 kV accelerating voltage.For XRD measurements, the nanocrystals were dried into solid powders and imported to a X-ray Diffractometer (Empyrean) with a temperature-controlled cell (TTK 450, Anton Paar GmbH).The diffraction patterns were acquired using Cu K  radiation operated at 40 kV and 40 mA, with a step width of 0.026°in the 2 range from 10°to 50°.
Transient Absorption Spectroscopy: TA measurements were performed on two femtosecond-amplifier-based transient spectrometers.For broadband TA covering ∼350−800 nm range, a regenerative amplified Ti:sapphire laser system (800 nm, 1 kHz, 70 fs pulse-duration; Coherent) and a TOPAS Optical Parametric Amplifier (OPA) were used.For hotcarrier relaxation measurements, a Pharos laser (1030 nm, 100 kHz, 230 fs pulse-duration; Light Conversion) and Orpheus-HP OPA were used.The details were provided in previous studies. [20,48]It is noted that directly monitoring hot-carrier relaxation at short wavelengths in TA spectroscopy is challenging due to the strong absorption of the current sample in that region.

Figure 1 .
Figure 1.Optical absorption and band structure of CsPbI 3 .a,b) Experimentally measured absorption spectrum (a) and transient absorption spectrum (b) of 9.0 nm CsPbI 3 nanocrystals dispersed in hexane at room temperature.c) Band structure of cubic CsPbI 3 computed from first principles with the HSE+SOC scheme.

Figure 2 .
Figure 2. Dynamical lattice distortion in pseudocubic CsPbI 3 .(a) High-resolution transmission electron microscope image of a typical CsPbI 3 nanocrystal, exhibiting clear octahedral tilting.(b) Evolution of the energy gaps of CsPbI 3 at high symmetry points Γ (0.0, 0.0, 0.0) and R (0.5, 0.5, 0.5) as a function of time.The blue line shows the accumulated average value of the bandgap at R (starting from 1000 fs).Four representative configurations (Config-1 − Config-4) are selected for more detailed analysis of the electronic structure.The initial atomic structure (ideal cubic perovskite) for the AIMD simulation is depicted.(c) X-ray diffraction pattern of CsPbI 3 nanocrystals at 303 K in comparison with those of the configurations extracted from AIMD simulations (from 0 to 2000 fs).

Figure 3 .
Figure 3. Electronic band structure of CsPbI 3 at room temperature.a-d) Atomic structures of the four representative configurations from the AIMD trajectory.e-h) The corresponding unfolded band structures of the four selected configurations at room temperature.The yellow lines depict the band structure of cubic CsPbI 3 at 0 K.The experimental values of ΔE c and ΔE v are provided in the plots for comparison and the intra-VB and intra-CB transition energies of cubic CsPbI 3 are marked with black arrows.

Figure 4 .
Figure 4. Impact of collective octahedral tilting on the electronic structure of CsPbI 3 .a) Structural transition of CsPbI 3 from the cubic phase to the tetragonal or orthorhombic phases through octahedral tilting.b,c) Variation of the energy levels as a function of a generalized configurational coordinate (Q) during b) the cubic-to-tetragonal and c) cubic-to-orthorhombic phase transitions.The dotted lines indicate the location of the tetragonal or orthorhombic phase; the tilting angles of the Pb-I octahedra are given.The VBM energy of cubic CsPbI 3 is chosen as the zero reference.(d,e) Variation of the bandgap (E g ), ΔE c , and ΔE v during the d) cubic-to-tetragonal and e) cubic-to-orthorhombic transitions.f) Schematic orbital diagram of the cubic, tetragonal and orthorhombic phases of CsPbI 3 .

Figure 5 .
Figure 5. Slow hot-hole cooling enabled by band renormalization.a) Schematic band structure and hot-hole relaxation in lattice-distorted CsPbI 3 .Under 405 nm excitation (purple arrow), the high-k hot hole can rapidly relax by emitting phonons; under 440 nm excitation (blue arrow), energy-momentum mismatch might result in relatively slow hot-hole relaxation between the two valence bands.b) Absorption spectrum of the CsPbI 3 nanocrystals (brown solid line) and the different pump wavelengths (colored arrows) used to observe hot-carrier cooling.c-e) Time-dependent transient absorption spectra normalized at their bleach maxima under c) 660 nm, d) 440 nm, and e) 405 nm excitations.The highlighted regions in the left panels are enlarged in the right panels.A progressive spectral narrowing on the blue side of the bleach is absent in (c), particularly obvious in (d), and less as obvious in (e).