A theoretical perspective of the ultrafast transient absorption dynamics of CsPbBr3

Abstract Transient absorption spectra (TAS) of lead halide perovskites can provide important insights into the nature of the photoexcited state dynamics of this prototypical class of materials. Here, we perform ground and excited state molecular dynamics (MD) simulations within a restricted open shell Kohn‐Sham (ROKS) approach in order to interpret the characteristic features of the TAS of CsPbBr3. Our results reveal that properties such as the finite temperature band gap, the Stokes shift, and therefore, also the TAS are strongly size‐dependent. Our TAS simulations show an early positive red‐shifted feature on the fs scale that can be explained by geometric relaxation in the excited state. As excited‐state processes can crucially affect the electronic properties of this class of photoactive materials, our observations are an important ingredient for further optimization of lead halide based optoelectronic devices.


| INTRODUCTION
Over the last years, organic-inorganic lead halide perovskites have attracted great attention because of their impressive photovoltaic performance and their superb photoluminescence properties, which also render them a suitable choice for making perovskite light-emitting diodes (LEDs). [1][2][3][4][5][6][7][8][9][10] Despite the rapid progress in device fabrication, a thorough understanding of the fundamental nature of the excited states of this class of materials is still missing. [11][12][13][14][15][16][17] Such characterization is an important ingredient for further optimization, as excitedstate processes can crucially affect the optoelectronic performance.
Among the different lead halide perovskites, [2][3][4][5]7 CsPbBr 3 is especially suited for investigations of the excited state dynamics via transient absorption spectroscopy experiments due to its higher photostability compared to other organic-inorganic lead halide perovskites. 11,[18][19][20][21][22][23][24][25][26] The large band gap of CsPbBr 3 that lies within the range of 2.25-2.36 eV according to measurements performed by different groups, 22,27 prevents direct photovoltaic application, but CsPbBr 3 is instead a promising material for LEDs and laser applications. 9,10,28,29 Although several femtosecond transient absorption spectroscopy measurements have been conducted for MAPbI 3 , MAPbBr 3 , and CsPbBr 3 , 14,15,17,23,24,[30][31][32] the interpretation of the mechanistic origin of the observed features is still under debate. More specifically, ultrafast transient absorption studies of cubic CsPbBr 3 nanocrystals as a function of size showed that for small nanocrystals of $4 nm edge length, strong quantum confinement effects are manifested in the excited-state dynamics that differ from the ones observed for the bulk material. 33 In addition to the size-dependent TAS, size-dependent absorption spectra and size-dependent Stokes shifts were revealed when photoluminescence measurements of CsPbBr 3 nanocrystals were performed. 11,33,34 In order to understand the origin of the Stokes shift in CsPbBr 3 nanocrystals, Brennan et al., 35 performed static density functional theory (DFT) calculations of the band structure, suggesting that for particles between 2 and 5 nm size, the observed shift is due to the existence of an intrinsic confined hole state 70-260 meV above the valence band edge which is relatively dark in absorption due to its low density of states, but bright in emission.
The time-dependent transient absorption measurements of CsPbBr 3 performed by Yarita et al. 17 revealed an early (0.4 ps) evolution of a positive transient absorption signal at 509 nm and a negative one at 528 nm. The latter signal disappeared at later time delays (≥7.5 ps) giving rise to a unique positive feature. Ultrafast transient absorption spectroscopy that was used to investigate the origin of the exceptionally high photoluminescence quantum yield of CsPbBr 3 quantum dots, 23 showed an exciton bleaching feature at $480 nm at early time delays (0.2-0.5 ps) and an exciton absorption feature at $495 nm at the same delay, which the authors assigned to state-filling-induced bleach and hot-exciton-induced red-shift, respectively.
After the hot-exciton relaxation (≥2 ps), the exciton-induced shift disappeared and the TAS was solely dominated by the exciton bleaching feature. 23 According to the authors, the high photoluminescence quantum yield of CsPbBr 3 quantum dots is a result of the negligible electron-and hole-trapping pathways.
On the other hand, Wei et al. 24

| RESULTS AND DISCUSSION
For the simulation of TAS, we considered an approach similar to the one followed in transient absorption experiments. More specifically, transient absorption spectroscopy is a time-resolved technique that allows monitoring the absorption spectrum of a sample after photoexcitation through a probe laser pulse at different delay times with respect to the pump pulse. In detail, when a sample is photoexcited, its absorption spectrum is modified due to the different distribution of the photoinduced charge carriers. 36 The transient absorption signal ΔA at a certain delay time τ is defined as the difference between the pumped A E, τ ð Þ and unpumped A E ð Þ signals, as described by The result of a transient absorption spectroscopy measurement is thus a differential absorption signal that is a function of both probe photon energy and time delay.
As a first test of the computational scheme, we checked if we can reproduce the experimentally observed Stokes shift of CsPbBr 3 . 11,33,34 For this purpose, we exploited the more expedient ROKS approach, a DFT-based method that describes the first nondegenerate excited singlet state of a system. 37 The absorption and emission energies at 0 K were calculated using the real-space equivalent of the fully k-point sampled system of the most stable phase of CsPbBr 3 at room temperature, 38 the orthorhombic phase. The experimental structure of CsPbBr 3 is shown in Figure 1. Benchmark calculations showed that such an appropriate real-space analog can be found with a supercell containing 160 atoms. In order to calculate the TAS, we exploited Equation (1)  Since spin-orbit coupling is known to be strong in lead halide perovskites, we also determined the analogous spectrum for triplet state. 39 The triplet state is determined by performing single point calculations for the same configurations that have been used in the calculation of the Kohn-Sham band gaps but using different spin multiplicity. As shown in Figure  In addition, we find that for this smaller size system temperature effects lead to a more pronounced blue-shift of the steady-state spectrum. In fact, the maximum of the band for the lowest singlet and trip- feature. 32 The absence of this feature following this approach further supports our suggestion that the early red-shifted feature is due to excited-state geometric relaxation. To further support our suggestion we examined the geometric distortions by comparing the Pb-Br distances of the ground-state trajectory at 300 K with the ones of the excited-state trajectories. We notice that the ground-state trajectory has only one set of similar Pb-Br distances, which average is 3.05 Å.
On the other hand, the Pb-Br distances extracted from the excitedstate dynamics trajectory can be classified into two different sets, which averages are 3.23 and 2.99 Å, accordingly.
We also simulated TAS performing two ROKS excited state dynamics simulations using the 160 atoms system. Our simulations ( Figure S2 in  have not been taken into account in view of the well-known fortuitous error cancellation between spin orbit coupling and many-body effects described e.g. by means of GW-self energy correction. 39,53,54 Quantum Espresso was chosen for the band gap calculations at 0 K, with a sampling of the Brillouin zone with a 7 Â 7 Â 7 Monkhorst-Pack shifted k-point grid. 55 To calculate the corresponding gap of the realspace sample, the CPMD code was used instead. In Quantum Espresso calculations, the interactions between valence electrons, core electrons and nuclei are described by ultrasoft pseudopotentials. 56 The Kohn-Sham orbitals are expanded in a plane wave basis set with a kinetic energy cutoff of 40 Ry and a density cutoff of 240 Ry. The above values have been chosen after having performed convergence tests for the total energy, the band gap, the pressure, the stresses and the atomic forces. For the calculations with CPMD, the wavefunction cutoff was set to 90 Ry and Goedecker normconserving pseudopotentials were employed. [57][58][59] Car-Parrinello MD simulations for the ground state in the NVT ensemble were performed using the CPMD code. 51 A time step of 5 a.

| CONCLUSIONS
u. was used with a fictitious mass parameter of 800 a.u. The temperature in the simulations was set to 300 K and was controlled by three Nosé-Hoover thermostats, (one for each species) with a coupling frequency of 1500 cm À1 . 60 instead. Finally, the convolution of the spectra was done by applying Gaussian functions, employing the comp_chem_py package. 63