Unravelling the Effects of A-Site Cations on Nonradiative Electron

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Unravelling the Effects of A Site Cations on Nonradiative Electron-Hole Recombination in Lead Bromide Perovskites: Time-Domain Ab Initio Analysis Jinlu He, Wei-Hai Fang, and Run Long J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b02115 • Publication Date (Web): 10 Aug 2018 Downloaded from http://pubs.acs.org on August 13, 2018

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The Journal of Physical Chemistry Letters

Unravelling the Effects of A Site Cations on Nonradiative Electron-Hole Recombination in Lead Bromide Perovskites: Time-Domain Ab Initio Analysis Jinlu He,1 Wei-Hai Fang1, Run Long1∗ 1

College of Chemistry, Key Laboratory of Theoretical & Computational

Photochemistry of Ministry of Education, Beijing Normal University, Beijing, 100875, P. R. China Abstract: Lead bromide perovskites APbBr3 (A = Cs, MA, FA) hold great promise in optoelectronics and photovoltaics. Because the bandgap of the three materials are similar, and also because the A site cation does not contribute to band edges, leading one to expect minor influence of A site cation on the excited-state lifetime of the perovskites. Experiments defy the expectation. By performing ab initio nonadiabatic (NA) molecular dynamics combined with time-domain density functional simulations, we demonstrate that the nonradiative electron-hole recombination times are on the order FAPbBr3 > MAPbBr3 > CsPbBr3, which are determined by the NA electron-phonon coupling because decoherence times are similar. The simulations show that the larger A site cation and the smaller NA due to larger A site cation suppressing the Pb-Br cages motions. The electron-hole recombination is slow, ranging from sub-nanosecond to nanoseconds, because the NA coupling is small, less than 3 meV, and because decoherence time is slow, less than 7 fs. Both the trend of recombination and the time scales show excellent agreement with experiments. The ∗

Corresponding author, E-mail: [email protected]

1

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time-domain atomistic simulations rationalize the experimental observations and advance our understanding of the cations on influencing perovskite excited-state lifetimes.

TOC only

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Hybrid inorganic-organic perovskites (HIOPs) solar cells have attracted intense attention due to rapidly growing photon-to-current conversion efficiencies increasing from 3.8%1 to 22.7%2 within eight years. The significant improvement of power conversion efficiencies originates from the excellent electronic and optical properties, including

low-trap

density,3

long

charge

carrier

lifetimes4

and

high

photoluminescence (PL) yields.5 However, HOIPs suffer from degradation under humidity and photo-irradiation and those issues weaken their photovoltaic properties.6 This drawback stimulates synthesizing all inorganic perovskites, such as cesium metal halides perovskites, which show better light and moisture stability.7 Similar to HOIPs, the all inorganic perovskites contain many excellent properties such as high fluorescence quantum yields,8 and long carrier diffusion lengths.9 These advanced properties make all inorganic perovskites are excellent candidates for a number of practical applications, including light-emitting diodes,10-13 optically pumped lasers,14-16 photocatalytic water-splitting assemblies,17-19 and solar cells.20-22 While the power conversion efficiencies of inorganic perovskites are lower than organic– inorganic hybrid halide perovskites7 stemming from the short excited state lifetimes.23 It is interesting because the bandgap of the APbBr3 (A =Cs, CH3NH3 (MA), HC(NH2)2 (FA)) is close to each other (CsPbBr3: 2.36 eV;24 MAPbBr3: 2.34 eV;25 FAPbBr3: 2.26 eV26), and also because the A site cation

does not directly contribute

to band edges. As a result, one expect that they should have an insignificant influence on the excited state lifetime of the APbBr3. 3



Recently, Jana et al. have demonstrated using time-resolved photoluminescence spectroscopy correlated with scanning electron microscopy that the average fluorescence lifetime of lead bromine perovskites APbBr3 are significantly different, in which the organic cations carry a fundamental advantage over the inorganic cation for achieving the long-lived excited state lifetime.23 One may wonder what the factors lead to the different excited state lifetimes in APbBr3 perovskites, because the band edges are entirely composed by lead and bromine orbitals.27 In order to explore the underlying mechanism responsible for the disparity in the excited state lifetime, first-principles atomistic time-domain simulations are needed to simulate the nonradiative electron-hole recombination in APbBr3, generating the mechanistic understanding on the excited state quantum dynamics, and providing valuable guidelines for materials design and further improvement. Previous first-principles ab initio quantum-classical dynamics simulations showed that the rearrangements of organic cations affect the electron-hole recombination in MAPbBr3 because they change the bandgap, modify the mixing between electron and hole wave function, and alter the phonon-induced quantum coherence time.28 Wang and coauthors demonstrated, using linear scaling ab initio methods, that random orientation of organic cations causes a long-range potential fluctuations in HOIPs, which spatially locates electron and hole in different positons, extending the charge-carrier lifetime.29 Similarly, the photoinduced deformation of inorganic Pb-Br sublattice also extends the charge-carrier lifetime by several orders of 4


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magnitude compared with the perfect crystal in all inorganic perovskite.30-31 Motivated by the recent experimental and theoretical works,23, 28-29, 32-34 we have performed ab initio time domain35 nonadiabatic molecular dynamics (NAMD)36 simulations in order to explore the atomistic mechanisms serving to the observed disparity in the nonradiative electron-hole recombination time scales among the three lead bromide perovskites. The calculations provide a mechanistic understanding of the influence of A site cation (Cs, MA, FA) on the excited state lifetimes and the perovskite solar cell performance. The obtained nonradiative electron−hole recombination times are ranging from sub-nanosecond to several nanoseconds, and both the trend in the recombination and the time scales show excellent agreement with available experimental data.32-34 Our calculations show that electron-hole recombination proceeding rapidly in CsPbBr3, followed by MAPbBr3 and FAPbBr3 can be attributed to the small magnitudes of NA coupling beucase decoherence times are similar. Despite cations (Cs, MA, FA) contribute neither to the highest occupied (HOMO) nor to the lowest unoccupied molecular orbital (LUMO), they influence the atomic motions of lead and bromine, which primarily constitute the LUMO and HOMO in lead bromine perovskites,27 corresponding to the initial and final states for electron-hole recombination in the three systems. The size of cation is on the order FA > MA > Cs, leading to suppression of vibrations of inorganic Pb-I sublattice on the same trend. As a result, the NA electron-phonon coupling has the inverse order, achieving the 5



short-lived charge-carrier lifetime in CsPbBr3, and extending significantly in MAPbBr3 and FAPbBr3. The charge-carrier lifetimes are long, because the NA coupling is small, below 3 meV, and the time of quantum coherence is short, below 7 fs. The simulations advance our understanding of the influence of A site cation on the excited dynamics in lead bromine perovskites, and provide valuable insights into enhancing the performance of lead bromide perovskites solar cells. The NAMD simulations are performed by the decoherence-induced surface hopping (DISH) approach36 implemented within the framework of time-dependent Kohn-Sham density functional framework.37-38 The faster and lighter electrons are described quantum mechanically, while the slower and heavier nuclei are treated classically.38-40 DISH incorporates decoherence correction into quantum dynamics and captures the nature of nuclear trajectory branching resulting in surface hops.36 Decoherence correction is needed here, because the decoherence (pure-dephasing) time is significantly shorter than the electron−hole recombination time, which occur on sub-nanosecond to several nanoseconds.32-34 The decoherence time is calculated as the pure-dephasing time in the optical response theory.41 DISH36 is implemented40 within the classical path approximation, which significantly reduces the computational cost, allowing one to use a predetermined ground state trajectory to propagate the nonequilibrium electron-vibrational dynamics. The approach has been applied to study photoexcitation dynamics in a variety of systems, 28, 31, 42-47 including quantum dots,42 two-dimensional (2D) ruddlesden-popper perovskites,43 perovskite 6


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passivated by water,44 containing grain boundaries,45 forming polarons28 and localized hole,31 and interfaced with TiO2.46-47 A detailed theoretical method can be found elsewhere.39-40 Geometry optimization, electronic structure, adiabatic MD, and NA coupling calculations were obtained by the Vienna ab initio simulation package (VASP)48. The generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE)49 functional and projector-augmented wave (PAW)50 pseudopotentials were used. A 400 eV plane-wave basis energy cutoff was employed for geometry optimization and electronic structure calculations. The structure optimization for both lattice constants and atomic coordinates with Γ-centered 4 × 4 × 4 Monkhorst−Pack mesh stopped until the residual forces on each atom were below 0.01 eV/Å.51 A much denser 8 × 8 × 8 mesh was used for density of states calculations. The van der Waals interactions were described by the Grimme DFT-D3 method.52-53 To model electron-hole recombination, a 2 × 2 × 2 supercell was constructed using the optimized cubic phase unit cells of CsPbBr3, MAPbBr3, and FAPbBr3 in the present study, containing 40 atoms in the CsPbBr3 and 96 atoms in the MAPbBr3 and FAPbBr3 respectively. After geometry optimization at 0 K, the three systems were heated to 300 K by repeated velocity rescaling. Then, 6 ps adiabatic MD trajectories were generated at Γ point in the microcanonical ensemble with a 1 fs time step. To simulate the nonradiative electron−hole recombination, the first 1000 initial conditions were sampled from the adiabatic MD trajectories to perform NAMD simulations with the PYXAID code.39-40 7



The bandgaps of all three systems under investigation were scaled to the experimental value for each material for study of electron-hole recombination.24-26

Figure 1. Simulation cell showing the optimized geometry (top panel) of (a) CsPbBr3, (b) MAPbBr3 and (c) FAPbBr3 and three corresponding representative geometries during the MD run at 300 K (bottom panel). At room temperature, the distortion of the inorganic Pb-I sublattice is on the order CsPbBr3 > MAPbBr3> FAPbBr3 because larger cation suppresses Pb and I motions more significantly. Here, the size of cation is on the order FA > MA > Cs, and the deformation of the Pb-I framework has the same order.

Figure 1 presents the optimized geometries of the CsPbBr3, MAPbBr3, and FAPbBr3 at 0 K (top panels), and snapshots of the corresponding structures taken from the MD trajectory at 300 K (bottom panels). The optimized lattice constants of the CsPbBr3, MAPbBr3, and FAPbBr3 systems are 5.900, 5.920, and 5.990 Å, agreeing well with experimental data of 5.870,54 5.919,55 5.99456 for each material 8


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respectively. The calculated averaged bond length of the Pb-Br in the corresponding material is 2.950, 2.974, and 3.002 Å respectively, also showing an excellent agreement with experiments, in which they are 2.960 Å,57 2.970 Å57 and 3.000 Å58 for CsPbBr3, MAPbBr3, and FAPbBr3 systems. At room temperature, the bond length increases to 3.087, 3.088, and 3.098 Å respectively. The increase in the Pb-Br bond length of the MAPbBr3 and FAPbBr3 is smaller than that of in the CsPbBr3 after heating at 300 K, because the larger A site organic cations suppress the deformation of inorganic sublattice, in particular for the FAPbBr3. The change in the Pb-Br bond lengths reflects the strong electron-vibrational coupling existing in the CsPbBr3, followed by MAPbI3 and FAPbBr3. This point can be further demonstrated by the lattice free volume, calculated by the unit cell volume subtracting the constituent ions volumes.59 In general, a larger free volume provides more space for atomic motions, enhancing electron-phonon coupling. The volume of CsPbBr3, MAPbBr3, and FAPbBr3 of the optimized unit cell is calculated to be 205.38, 207.47, 214.92 Å3. The ionic radii of Cs+, MA+, FA+, Pb2+, and Br- are 1.88, 2.17, 2.53, 1.16, and 1.96 Å, respectively.60-61 Each APbBr3 unit cell contains one A (Cs+, MA+, FA+), one Pb2+, and three Br-, adding the spheres volumes in the unit cell gives the total ion volumes of 128.99 Å3 for CsPbBr3, 143.96 Å3 for MAPbBr3, and 168.99 Å3 for FAPbBr3. Therefore, the free volumes for the bromine and lead in the CsPbBr3, MAPbBr3, and FAPbBr3 are 76.39, 63.51, 45.93 Å3, respectively. The ratio between the free volumes and the total volumes denotes the specific free volumes,62 which are 37.19%, 30.61%, 9



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and 21.37% for the perovskites containing Cs, MA, and FA, respectively. The smaller specific free volume indicates that the amplitude of ion motions is smaller and the electron-phonon coupling is weaker. Hence, it is expected that the electron-phonon coupling decreases in the sequence CsPbBr3 > MAPbBr3 > FAPbBr3. In addition to the reported average bond lengths and specific free volumes, we calculate the canonically averaged standard deviation of the position of each atoms

, = − 〈 〉 , where represents the location of atom at time t along the 6 ps MD trajectories. A smaller value of standard deviation reflects smaller atomic fluctuation and weaker electron-phonon coupling. The standard deviations in the positions of Pb and Br atoms are calculated for the three systems, because the charge densities of HOMO and LUMO are primarily consisted by Pb and Br orbitals. Shown in Table 1, the standard deviations in the positions of Br/Pb atoms decrease in the following order: CsPbBr3 > MAPbBr3 > FAPbBr3. The decrease in atomic fluctuations reflects electron-phonon couplings decreasing in the same order. Thus, the analysis of bond length, specific free volume, and atomic fluctuation agrees with each other and confirms that electron-phonon coupling is strongest in the CsPbBr3, with reduction in the MAPbBr3 and FAPbBr3. Table 1. Standard Deviations in the Positions of Pb and Br Atoms in CsPbBr3, MAPbBr3 and FAPbBr3.

Pb

CsPbBr3

MAPbBr3

FAPbBr3

0.343

0.332

0.258

10


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Br

0.603

0.440

0.412

Figure 2 shows the projected density of states (PDOS) of the CsPbBr3, MAPbBr3 and FAPbBr3 systems, calculated at 0 K using the optimized geometries. The PDOS illustrates that the HOMO is primarily composed by the Br atoms, with secondary contributions of Pb atoms, and the LUMO arises primarily from Pb atoms, with small contributions of Br atoms, agreeing with previous works.

27, 63

The two key orbitals

constitute the initial and final states for electron-hole recombination in the three systems under investigation. Although the cations Cs+, MA+, and FA+ do not contribute to the band edges, they do affect the local inorganic Pb-Br structure via electrostatic interactions. The calculated direct bandgaps of CsPbBr3, MAPbBr3 and FAPbBr3 are 1.60 eV, 1.94 eV and 1.91 eV at the Γ point respectively, showing a good agreement with the bandgap for each material obtained using the same PBE functional.64-65 Due to well-known DFT problem,66 the obtained values are smaller than the experimental bandgap for CsPbBr3 (2.36 eV),24 MAPbBr3 (2.34 eV),25 and FAPbBr3 (2.26 eV).26 To directly mimic the experiments,23 we scaled the calculated bandgaps to their experimental values for simulation of electron-hole recombination in these materials. The influence of bandgap on the recombination time is small, because the recombination rate depends linearly on the bandgap according to gap law,67 and also because the difference in bandgaps and decoherence times (Table 2) of the three materials is small. Consequently, the electron-hole recombination times are 11



determined by the NA electron-phonon coupling.

Figure 2. Projected density of states (PDOS) of (a) CsPbBr3, (b) MAPbBr3, and (c) FAPbBr3 systems, split into contributions from the Pb, Br, and A cation. In all three systems, cation A contributes neither to the LUMO nor to the HOMO.

The NA matrix element relies significantly on both the HOMO/LUMO mixing −iħϕ ϕ and nuclear velocity d/dt . A larger mixing of the corresponding charge densities and faster nuclear dynamics typically generate stronger NA electron-phonon coupling. Shown in Figure 3 a-c, the HOMO is localized primarily on Br atoms, while the LUMO is localized mainly on Pb atoms. The wave function mixing is similar in all of the three perovskites. The difference in NA electron-phonon coupling stems from the nuclear velocity, Table 1. Because FA is larger than MA and Cs, and hence, the strength of the NA electron-phonon coupling decreases from CsPbBr3, MAPbBr3, and to FAPbBr3.

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Figure 3. Charge densities of the HOMO and LUMO. (a) CsPbBr3, (b) MAPbBr3 and (c) FAPbBr3 systems.

In order to characterize the phonons that couple to the electronic transition, we computed Fourier transforms (FT) of autocorrelation functions (ACF) of the HOMO−LUMO bandgap fluctuations in the three systems. Figure 4 shows that low-frequency vibrations in the 30−300 cm−1 region govern the nonradiative relaxation in all the three systems. Figure 4a shows in the CsPbBr3 system that the major peak at 67 cm-1 and the secondary peak at 133 cm−1 can be attributed to the vibrations of the inorganic Pb-Br octahedron and motion of Cs+ cations, respectively.68 Figure 4b demonstrates in the MAPbBr3 that the dominant peak at 33 cm-1 can be assigned to the Pb-Br inorganic framework distortion.69 The peaks at 133 cm-1 and 233 cm-1 can be assigned to the lurching of MA+ cations70 and torsional mode of the MA cations.71 For Figure 4c illustrates in the FAPbBr3 that the primary peak at 33 cm-1 can be also assigned to the Pb-Br sublattice distortion.69 The peak at 121 cm-1 can be ascribed to the vibration of 13



FA+ cations.72 The dominant vibrational mode creates the largest NA electron-phonon coupling and induces rapid loss of quantum coherence. Compared the spectral density of CsPbBr3 with the MAPbBr3 and FAPbBr3, the main peak in the CsPbBr3 has higher frequency because the size of Cs+ cations are smaller the MA+ and FA+ cations, which allow inorganic Pb-Br framework move more in space in CsPbBr3 than that in MAPbBr3 and FAPbBr3 and generate the strongest NA electron-phonon coupling and shortest decoherence time. The range of active vibrations in MAPbBr3 is wider than those in FAPbBr3, leading to stronger NA electron-phonon coupling and shorter decoherence time. The phonon modes originating from A site cation affect the interaction between themselves and the inorganic lead and bromide atoms via electric filed they created, affecting the electron-phonon coupling and quantum decoherence time. Overall, the NA electron-coupling decreases in perovskite containing Cs, MA, and FA, and the docoherence time decreases in the reverse order, Table 2. One should be noted that decoherence time of the three systems are similar and have insignificant influence on the difference of the charge recombination times.

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Figure 4. Spectral densities obtained by Fourier transforms of the autocorrelation functions, inset of Figure 5, for the fluctuations of the HOMO−LUMO energy gap in (a) CsPbBr3, (b) MAPbBr3, and (c) FAPbBr3.

The pure-dephasing functions computed using the second-order cumulant approximation in optical response theory41 are shown in Figure 5. The pure-dephasing times, τ, were obtained by fitting the pure-dephasing functions with a Gaussian, !" −0.5&/' . The pure-dephasing times decrease from FAPbBr3, to the MAPbBr3, to the CsPbBr3, Table 2, corresponding to 6.64, 6.19 and 5.73 fs, respectively. The moderate drop of dephasing time from FAPbBr3 to CsPbBr3 can be attributed to the different phonon modes discussed above. The un-ACF, calculated from the fluctuations of the HOMO-LUMO energy gap, inset of Figure 5 further gives the origin of pure-dephasing times differing each other. Generally, the larger initial value of the un-ACF and the shorter dephasing time, because under cumulative approximation, the initial value of the un-ACF equals to the variance of the energy gap, reflecting the strength of electron-phonon coupling. As a result, the pure 15



dephasing times of three systems decrease as the sequence: FAPbBr3 > MAPbBr3 > CsPbBr3. It is necessary to incorporate decoherence corrections into NAMD simulations because the 5-7 fs pure-dephasing times are extremely shorter than the sub-nanosecond to several nanoseconds electron−hole recombination times reported experimentally.32-34

Figure 5. Pure-dephasing functions for the LUMO−HOMO transition in CsPbBr3, MAPbBr3, and FAPbBr3. The timescales obtained by Gaussian fits are summarized in Table 1. The inset shows the un-ACF, whose initial values are equal to the bandgap fluctuation squared. In general, The bigger initial value results in the faster dephasing.41

Figure 6 shows the evolution of the population of the first excited state in the three perovskite systems. The nonradiative electron-hole recombination corresponds to a single-molecule process and decay exponentially. Calculation of the whole charge recombination dynamics up to tens of nanoseconds demands heavily computational 16


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cost, and thus we ran 20 ps NAMD simulations, in which the populations decay linearly due to the limited range, and extrapolated the recombination times based on the short dynamics. Shown in Table 2, the times, ', are obtained with the short-time linear approximation to the exponential decay, (& = !"−&/' = 1 − −&/', which are 0.88, 1.28, and 3.11 ns for CsPbBr3, MAPbBr3 and FAPbBr3 respectively, agreeing well with the data reported experimentally.32-34 Generally, pure DFT functional such as PBE used here, underestimates the band gaps of all the lead perovskites, and thus enhances the NA electron-phonon coupling because small band gap favors electron and hole wave functions mixing, leading to a faster electron-hole recombination. As discussed above, the difference in electron-hole recombination times for the three perovskites are primarily stemming from the NA electron-phonon coupling, because the decoherence times of three system are similar. The NA coupling is in the order: CsPbBr3 > MAPbBr3 > FAPbBr3, the recombination as result of same trend. The research demonstrates that A site cation plays a key role in determining the excited state lifetime of lead bromide perovskites through mainly influencing the NA electron-phonon coupling, suggesting that rational choice of A site cation can significantly reduce charge and energy losses and increase the conversion efficiency of perovskite solar cells.

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Figure 6. Electron-hole recombination dynamics in CsPbBr3, MAPbBr3 and FAPbBr3.

Table 2 Experimental Bandgap, Averaged Absolute NA Coupling, Pure-Dephasing Time, and Nonradiative Electron-Hole Recombination Time for CsPbBr3, MAPbBr3 and FAPbBr3. Bandgap

NA coupling

Dephasing

Recombination

(eV)

(meV)

(fs)

(ns)

CsPbBr3

2.36

2.28

5.73

0.88

MAPbBr3

2.34

1.62

6.19

1.27

FAPbBr3

2.26

1.01

6.64

3.10

In summary, we have investigated the nonradiative electron−hole recombination in CsPbBr3, MAPbBr3 and FAPbBr3 perovskites by NAMD combined with TDDFT, in order to resolve the microscopic mechanism serving to the difference in excited state lifetimes of the three materials. The simulations support the experimental 18


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observations that perovskite containing larger A site cation favors longer excited state lifetime. The ionic radii are in the sequence: FA > MA > Cs, and as a result of the excited state lifetimes follow the same trend. This is surprising because the bandgaps and dephasing times in the three materials are similar. The simulations demonstrate that A site cations steer the atomic fluctuations of inorganic Pb-Br sublattice and affect the NA coupling although A site cations do not directly contribute to the band edge states. The larger the A site cation and the smaller atomic motions in Pb and Br, which reduces NA electron-phonon coupling and leads the recombination to proceed slow in perovskite containing FA, moderate in MA, and fast in Cs. The reported simulations provide a detailed description of the complex quantum dynamics in the lead bromine perovskites, generating important insights and suggesting design principles for improve perovskite solar cells performance via a rational choice of cations.

Acknowledgements This work was supported by the National Science Foundation of China, Grant Nos. 21573022, 51861135101, 21520102005, and 21421003. R. L. is grateful to the Fundamental Research Funds for the Central Universities, the Recruitment Program of Global Youth Experts of China, and the Beijing Normal University Startup.

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