Time-Domain ab Initio Analysis Rationalizes the Unusual Temperature

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Time-Domain Ab Initio Analysis Rationalizes the Unusual Temperature Dependence of Charge Carrier Relaxation in Lead Halide Perovskite Wei Li, Jianfeng Tang, David Casanova, and Oleg V. Prezhdo ACS Energy Lett., Just Accepted Manuscript • DOI: 10.1021/acsenergylett.8b01608 • Publication Date (Web): 10 Oct 2018 Downloaded from http://pubs.acs.org on October 10, 2018

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ACS Energy Letters

Time-Domain Ab Initio Analysis Rationalizes the Unusual Temperature Dependence of Charge Carrier Relaxation in Lead Halide Perovskite

Wei Li,1* Jianfeng Tang,1 David Casanova,2,3 Oleg V. Prezhdo4*

1College

of Science, Hunan Agricultural University, Changsha 410128, People’s

Republic of China 2Kimika

Fakultatea, Euskal Herriko Unibertsitatea and Donostia International Physics

Center 20018 Donostia, Euskadi, Spain 3IKERBASQUE,

4Department

Basque Foundation for Science, 48013 Bilbao, Euskadi, Spain

of Chemistry, University of Southern California, Los Angeles, CA 90089,

United States

Corresponding authors: E-mail: [email protected] (W.L.) E-mail: [email protected] (O.V.P.) 1 ACS Paragon Plus Environment

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Abstract Increased charge carrier lifetimes at elevated temperatures constitute an intriguing and valuable feature of hybrid organic-inorganic perovskites that are subject to heating in solar energy applications. We rationalize this peculiar behavior at the atomistic level using real time time-dependent density functional theory and nonadiabatic molecular dynamics. Focusing on tetragonal MAPbI3, we demonstrate that temperature has different effects on the organic and inorganic subsystems, and leads to subtle structural changes, decreasing nonradiative electron-hole recombination. First, charge-phonon interactions are decreased because the libration dynamics of the organic component at higher temperature reduces the oscillation amplitude of the Pb-I lattice that supports electrons and holes. Second, thermal disorder localizes wavefunctions, reducing the nonadiabatic charge-phonon coupling. Third, tilting of the inorganic octahedra increases the bandgap, extending charge carrier lifetime further. The detailed timedomain atomistic analysis of the uncommon dependence of charge recombination on temperature emphasizes the key role played by the organic cation, establishes important structure-property relationships, provides valuable insights into efficient performance of perovskites solar cells, and highlights factors required to maintain and improve such performance.



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Hybrid organic-inorganic perovskites, including MAPbX3, X=Cl, Br, I, have attracted strong interest for photovoltaic applications due to their outstanding performance.1–8 The power conversion efficiency of perovskite solar cells (PSCs) reached 23.2% in 2018.1 Perovskite materials exhibit unique physical properties, their chemical composition and bandgap are easily tunable, and they exhibit large charge carrier mobilities, absorption cross sections and diffusion lengths.5 Long charge carrier lifetimes, ranging from few to hundreds of nanoseconds, have been reported for hybrid perovskites.9,10 Explanations for such long carrier lifetimes include unusual defect tolerance,11–14 formation of polarons,15,16 direct-to-indirect bandgap transition,17,18 fast carrier relaxation enabled by spin-orbit interactions,19 Rashba splitting,20,21 and ferroelectric domains.22 Most of these perovskite features are believed to correlate with the unique organic-inorganic hybrid perovskite structure. Although significant progress has been achieved with PSCs, the recorded efficiencies are still below the theoretical maximum.23 Nonradiative electron-hole recombination reduces charge carrier lifetimes, and constitutes a major pathway for charge and energy losses. To improve the efficiency further, a fundamental understanding of the factors limiting PSCs performance is needed. PSCs devices are subject to significant temperature fluctuations under normal operating conditions.24,25 Therefore, the impact of temperature on charge carrier lifetime is of fundamental interest to the community.15,25–29 Higher temperature



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increases thermal motions, and leads to larger nonadiabatic (NA) electron-phonon coupling30,31 and decreased charge carrier lifetimes in conventional semiconductors and metals.32,33 However, the situation is different in the hybrid organic-inorganic perovskites.

Using

photoluminescence

spectroscopy,

Herz

and

co-workers

demonstrated that the bimolecular rate constant for charge carrier recombination in MAPbI3 decreases with rising temperature.25 Kanatzidis and co-workers found a similar observation in the lead halide perovskite.34 Since PSCs are subject to heating, and its efficiency depends on charge carrier lifetimes, it is essential to understand the origin of the uncommon and advantageous influence of temperature on the charge carrier dynamics. Further, MAPbI3 perovskite was shown to exhibit an unusual positive temperature dependence of the bandgap. A blue-shift of the photoluminescence peak, up to tens of meV, was observed in MAPbI3.25,35–38 This is also in contrast with the conventional semiconductors in which the bandgap decreases with increasing temperature.39,40 One expects that temperature variation should have a significant effect on motions of organic cations, which display complex anharmonic dynamics, including fast reorientations.41 In turn, organic cations can influence dynamics of the inorganic lattice. It is challenging to elucidate the complex interplay of electronic and nuclear dynamics in hybrid perovskites from experimental data only, motivating detailed atomistic simulations of the structure-property relationships in PSCs. In this letter, we rationalize theoretically the unusual and favorable temperature dependence of charge carrier lifetimes in a hybrid organic-inorganic perovskite. 6 ACS Paragon Plus Environment

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Motivated by the recent experiments,15,25,34,42 we perform ab initio NA molecular dynamics (MD) simulations of nonradiative electron-hole recombination in perovskite MAPbI3, Figure 1, at different temperatures. We focus on tetragonal MAPbI3, which has become an extremely popular material for high performing PSCs. The simulations show that increased temperature slows down the recombination, rationalizing the experimental observation25 and providing a detailed mechanism of the uncommon effect. Namely, an increased temperature enhances motions of the organic MA cations, but suppresses vibrations of the inorganic Pb-I lattice that supports charge carriers. Thermal fluctuations localize electron and hole wavefunctions, while subtle structural changes increase the bandgap. All these factors decrease the NA coupling and extend charge carrier lifetimes. The atomistic insights into the influence of temperature on charge carrier relaxation in hybrid organic-inorganic perovskites, provided by the simulations, generate fundamental knowledge and suggest design principles for maintaining and improving performance of PSCs that operate at ambient and elevated temperatures. The NA-MD simulations are carried out using the semiclassical decoherence induced surface hopping (DISH) approach43 implemented within with real-time timedependent density functional theory (TDDFT).44 The lighter electrons are treated quantum-mechanically, whereas the heavier nuclei are treated semi-classically. The DISH-TDDFT approach has been applied successfully to investigate charge carrier relaxation in a wide range of systems.14,22,45–56 More details regarding this approach can 7 ACS Paragon Plus Environment


be found elsewhere.44,57 The ground state geometry optimization and electronic structure calculations are performed using the VASP code58 employing the PerdewBurke-Ernzerhof (PBE) exchange-correlation functional and projector augmented wave (PAW) pseudopotentials with an energy cutoff of 500 eV. Starting with the bulk tetragonal structure of MAPbI3 reported by Walsh and coworkers59 based on the crystallographic information, we optimize the bulk structure further until the force on each atom is less than 0.01 eV/Å. The geometry optimization is followed by 2 ps BornOppenheimer MD simulation using repeated velocity rescaling, to bring the temperature from 0 K to 200 K and 300 K. Then, 3 ps microcanonical BornOppenheimer MD trajectories are obtained at each temperature with a 1 fs time step, and are used for the subsequent NA-MD simulations. The trajectories provide good representation of the atomic motions responsible for the experimentally observed temperature dependence of charge carrier dynamics.25 The state energies and NA couplings computed along the trajectories are repeated four times to obtain 12 ps NA Hamiltonians. The DISH calculations are performed under the classical path approximation44,57 by averaging over 2000 initial geometries and 1000 stochastic realizations of the DISH process for each geometry. A 4x4x4 Monkhorst-Pack mesh is used for the structure optimization and MD simulations. Spin-orbit coupling (SOC) is significant in lead halide perovskite due to the presence of heavy elements such as Pb and I.60 However, SOC is not included in the current work since it does not change the structure,61 and because PBE-DFT without SOC provides the bandgap that matches 8 ACS Paragon Plus Environment

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ACS Energy Letters

experiment due to error cancellation.60,62 The NA-MD simulations are performed using the PYXAID code.44,57 First, consider how temperature variation perturbs perovskite structure. Hybrid perovskites are quite “soft” and exhibit a variety of motions, many of which are highly anharmonic.7,63,64 The inorganic lattice vibrates with a large amplitude, and the organic cations librate and, on a longer timescale, rotate. Because electrons and holes near conduction band minimum (CBM) and valence band maximum (VBM) are localized on the inorganic lattice, lattice fluctuations are of primary importance for charge recombination. Thermal fluctuations increase the average tilting and fluctuations of the inorganic octahedra. Figure 2 shows that the Pb-I-Pb bond angle fluctuates by nearly 40°. The fluctuation amplitude is larger at the higher temperature, as one can expect. The average Pb-I-Pb bond angle is smaller at 300 K than 200 K, 157.6° vs. 154.6°, Figure 2 and Table 1. Evolution of the Pb-I-Pb bond angle correlates with changes in the VBM-CBM bandgap, Figure 2. Generally, the bandgap increases when the Pb-I-Pb bond angle decreases and octahedral tilting increases, as found in the previous experimental and computational investigations.65–68 The gap also increases due to downshift in VBM, Figure S1, which is largely based on the antibonding orbital that shifts down in energy due to reduced orbital splitting as atoms move farther apart at the higher temperature.35,69 The canonically averaged bandgap is 1.68 eV at 200 K and 1.80 eV at 300 K, in agreement with the previous report.35 9 ACS Paragon Plus Environment


Since the NA electron-phonon coupling depends on changes in the electron and hole wavefunction due to nuclear motions, we characterize the magnitude of atomic motions by computing the canonically averaged root-mean-square velocity of each atom. A larger velocity indicates stronger fluctuations, leading one to expect a stronger NA coupling and a faster charge recombination. Table 2 shows the velocity averaged over all atoms, and separately over atoms belonging to the organic MA cations and the inorganic Pb-I lattice. Considering all atoms, we see a larger fluctuation at the higher temperature, as expected. The key observation enters when one studies separately fluctuations of the organic and inorganic atoms. The atoms of the organic MA cations move more at 300 K; however, surprisingly, the atoms of the inorganic Pb-I lattice fluctuate less at 300 K, Table 2. The result is critical, because it is the motion of the Pb and I atoms that determines the NA coupling. The simulations show that librations of the organic cations accelerate at the increased temperature and suppress vibrations of the inorganic Pb-I lattice.


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ACS Energy Letters

Figure 1. Schematic representation of nonradiative electron-hole recombination investigated in this work (left). Optimized ground state structure of the MAPbI3 perovskite in the tetragonal phase (right).

Figure 2. Evolution of (a) Pb-I-Pb bond angle and (b) bandgap in the MAPbI3 perovskite at 200 K and 300 K. The dashed lines denote the canonically averaged values. The smaller Pb-I-Pb bond angle at the higher temperature demonstrates larger octahedron tilting and correlates with the wider VBM-CBM energy gap, Table 1, in agreement with



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experiment.37 The bandgap fluctuates more at the lower temperature, indicating stronger electron-phonon coupling. Table 1. Canonically averaged Pb-I-Pb bond angle (θ), bandgap (Eg), standard deviation of bandgap fluctuation (σE), absolute NA coupling (NAC), pure-dephasing time (T2∗ ), and nonradiative relaxation time (T1) for charge recombination in MAPbI3 at 200 K and 300 K. θ

Eg

σE

NAC

(deg.)

(eV)

(eV)

(meV)

(fs)

(ns)

200 K

157.6

1.68

0.12

0.82

5.8

5.5

300 K

154.6

1.80

0.10

0.69

6.0

12.0

T2∗

T1

Table 2. Root-mean-square velocity (Å/fs) of atoms in MAPbI3 at 200 K and 300 K. totala

MAb

Pb-Ic

200 K

0.023

0.025

0.019

300 K

0.026

0.032

0.014

a

averaged over all atoms

b

average over atoms in MA

c

average over Pb and I atoms

In addition to the sensitivity of the wavefunctions to atomic motions, the NA coupling depends on localization and overlap of electron and hole wavefunctions. Figure 3 shows charge densities of the VBM and CBM states for the optimized structure and representative structures as 200 K and 300 K. VBM is mainly composed of I-p atomic orbitals that couple to the Pb-s orbitals in an antibonding way, and CBM


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originates from nonbonding Pb-p orbitals.70 Both VBM and CBM delocalize over the entire cell of the optimized structure. Thermal fluctuations perturb system symmetry and induce orbital localization. For example, the VBM is localized within the middle inorganic layer in the 200 K example shown in Figure 3b. The states become more localized, and the VBM-CBM orbital overlap decreases at 300 K. In the example shown in Figure 3c, the VBM is localized in the middle inorganic layer, while the CBM is supported by the other the layers. Generally, thermal atomic fluctuations localize VB states more than CB states. This is because VB arises from atomic orbitals of I atoms which are lighter than Pb atoms and fluctuate more. Also, iodine is a non-metal and tends to form directional covalent bonds that are perturbed more strongly by fluctuations than less directional bonds of the Pb metal. In order to characterize the overlap between electron and hole, we calculated the overlap of electron and hole charge densities for 300 structures chosen randomly along the MD trajectory, Figure S2. Note that we compute overlap of charge densities, i.e. wavefunctions squared, and not wavefunctions, since the latter are orthogonal. The averaged absolute values of the NA couplings, reported in Table 1, correlate with the extent of the overlap between the VBM and CBM charge densities. The coupling is weaker at 300 K (0.69 meV) than 200 K (0.82 meV), because the VBM and CBM orbitals are more localized and overlap less, Figures 3 and S2, and since motions of the Pb and I atoms are suppressed by motions of MA, Table 2.



Figure 3. VBM and CBM charge densities for (a) the optimized ground state structure, and representative snapshots at (b) 200 K and (c) 300 K. VBM originates primarily from antibonding coupling between Pb-s and I-p atomic orbitals, while CBM is supported by nonbonding Pb-p orbitals. Both VBM and CBM charge densities are spread over the entire simulation cell of the optimized structure, and become more localized as temperature increases.

Fourier transforms (FT) of phonon-induced fluctuations of the VBM-CBM energy gaps characterize the phonon modes that participate in the nonradiative electron-hole recombination, accommodate the dissipated energy, and induce loss of quantum coherence within the electronic subsystem. FTs of the gap autocorrelation functions (ACF), known as spectral densities or influence spectra, are reported in Figure 4a. The data show that only low frequency phonon modes are involved in the 14 ACS Paragon Plus Environment

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nonradiative relaxation process, in agreement with the previous investigations.14,46 Higher frequency modes contribute little to the spectrum density, suggesting that the organic cations do not influence the charge recombination directly. The 200 K system shows a strong peak around 100 cm-1 and a weaker peak around 50 cm-1. These peaks can be assigned to Pb-I-Pb bending and stretching modes, respectively, as determined by Raman spectroscopy.71 In comparison, the 300 K system shows multiple lower peaks over a broader frequency range. This is an indication of a structure perturbation caused by temperature. The phonon mode around 200 cm-1 can be attributed to librations of the MA cations, which contribute to the nonradiative electron-hole recombination between the CBM and VBM by coupling to the Pb-I inorganic lattice. The peak intensities in the phonon influence spectrum decrease at the higher temperature, which is unusual for semiconductors.32,72 This happens in the hybrid organic-inorganic perovskite, because MA motions grow in amplitude and constrain motions of the Pb-I lattice, which couples to the electronic subsystem. The temperature induced change in the influence spectrum, Figure 4a, is in agreement with the root-mean-square velocity data shown in Table 2. In particular, as the temperature increases, the peak intensity of the mode around 200 cm-1 grows, while the peaks around 100 cm-1 are suppressed, Figure 4a. Based on the mode assignment, this result is interpreted as enhancement of vibrations of the inorganic lattice and suppression of motions the organic species. Correspondingly, the root-mean-square



velocity of MA increases and the root-mean-square velocity of Pb and I decreases with increasing temperature, Table 2. The NA coupling, responsible for the nonradiative charge recombination, is sensitive to both atom velocity and wavefunction overlap, and these two factors are not necessarily correlated. In the current case, increased temperature increases disorder, leading to decreased overlap, Figures 3 and S2. The disorder correlates with larger fluctuations of all atoms, Table 2. At the same time, the perovskite structure is such that larger fluctuations of the organic species limit fluctuations of the inorganic lattice, as reflected in the reduced root-mean-square velocity of the Pb and I atoms, Table 2. It is this unusual anti-correlation that explains the unexpected observed temperature dependence of charge carrier lifetimes. The pure-dephasing functions, reflecting loss of quantum coherence in the electronic subsystem due to coupling to phonons, can be calculated using the secondorder cumulant approximation of the optical response theory.73–76 They are shown in Figure 4b. Pure-dephasing represents elastic electron-phonon scattering. The puredephasing times, τ, are obtained by Gaussian fitting of the pure-dephasing functions, exp[−0.5(t/τ)2], and are listed in Table 1. The pure-dephasing/decoherence times are generally very short, 5.8 fs and 6.0 fs for 200 K and 300 K, respectively. It is unusual that pure-dephasing is faster at a lower temperature. The inverse of the pure-dephasing time determines homogeneous optical linewidth, and it is common that optical lines become sharper at lower temperatures, corresponding to longer pure-dephasing.77 The 16 ACS Paragon Plus Environment

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insert in Figure 4b presents un-normalized ACFs (u-ACFs) of the phonon-induced fluctuations of the VBM-CBM energy gaps. u-ACFs can provide useful information on the pure-dephasing process. Generally, the dephasing is fast if the energy gap fluctuation, represented by the initial u-ACF value, is large, and if correlations in gap fluctuations decay rapidly.78 A large fluctuation corresponds to large signal magnitudes in the influence spectrum, Figure 4a. The decay is rapid if the fluctuation is induced by multiple phonon modes. The u-ACFs shown in the insert of Figure 4b decay on similar timescales, while the initial value is larger at 200 K than 300 K, rationalizing the faster pure-dephasing, Table 1.

Figure 4. (a) Spectral densities and (b) pure-dephasing functions for MAPbI3 at 200 K (black line) and 300 K (red line). The insert in part b shows un-normalized autocorrelation functions of the VBM-CBM energy gap fluctuations. Square roots of the initial values correspond to the standard deviations of gap fluctuations listed in Table 1. The lower frequency signals are stronger and higher frequency signals are weaker at 200 K, because MA motions are suppressed and Pb-I motions are enhanced, Table 2.



Figure 5 shows decay of the population of the first excited state due to nonradiative electron-hole recombination in MAPbI3 at 200 K and 300 K. The recombination time reported in Table 1 are obtained by fitting the data to the shorttime, linear approximation to the exponential decay, f(t) = exp(−t/τ) ≈ 1 − t/τ. The simulated carrier lifetime fall within the range of the experimentally reported values9,10 and demonstrate that increased temperatures reduce nonradiative electron-hole recombination and extend charge carrier lifetimes, rationalizing the observed temperature dependence.25 The difference in the charge recombination rates can be rationalized by temperature induced changes in the electronic energy gap, NA coupling magnitude, and pure-dephasing time, see Table 1. Generally, the recombination is slow if the electronic energy gap is large, the NA coupling is weak, and quantum coherence is short-lived. These trends can be rationalized with the Marcus theory79 or Fermi’s golden rule.80 Since the absolute value of the gap is large, compared to the reorganization energy, the system is in the inverted Marcus regime, and larger gaps decrease the rate. The NA coupling determines the pre-exponential factor of the Marcus rate formula, and is the source of coupling between the initial and final states during the nonradiative relaxation. Larger NA couplings increase the rate. The coherence time is directly related to reorganization energy, as well as to the FranckCondon74 and Huang-Rhys81 factors. The limit of infinitely fast decoherence leads to the quantum Zeno effect,82 which demonstrates that shorter coherence leads to slower 18 ACS Paragon Plus Environment

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rates. In the current case, both NA coupling and energy gap favor faster relaxation at 200 K, Table 1. The slightly shorter coherence time is less important. The atomistic origins of the temperature induced differences in the energy gap, NA coupling and coherence times have been analyzed in detail above. The analysis rationalizes the experimental observation25 on the unusual temperature dependence of charge carrier lifetimes and advances our understanding of structure-property relationships in lead halide perovskites.

Figure 5. Excited state population decay in MAPbI3 at 200 K and 300 K. Charge recombination slows down at the increased temperature, in agreement with experiment,25 due to decreased NA electron-phonon coupling, Table 1. The coupling decreases at 300 K because the VBM and CBM orbitals become more localized, Figure 3, and motions of the Pb and I atoms are suppressed, Table 2.



In summary, we have rationalized the unusual temperature dependence of charge carrier lifetimes the MAPbI3 perovskite, using NA-MD combined with ab initio real-time TDDFT. Observed experimentally, the longer lifetimes of charge carriers at higher temperatures are favorable for solar energy applications, in which materials undergo substantial heating. The perovskite behavior is opposite to the trend seen with the majority of semiconductors, in which electron-phonon interactions and nonradiative charge recombination increase with temperature. The mechanism underlying the unusual behavior involves an interplay between motions of the inorganic and organic components of the hybrid perovskite, leading to subtle structural rearrangements and weakened charge-phonon interactions. In particular, thermal motions of MA cations induce a tilting of the inorganic octahedra and suppress Pb-I vibrations that promote nonradiative electron-hole recombination. The recombination is reduced further by thermally induced localization of electron and hole wavefunctions, decreasing the NA coupling. In addition, the structural changes increase the bandgap, which also favors a longer excited state lifetime. Increased temperatures reduce not only inelastic electron-phonon scattering, but elastic scattering as well. The pure-dephasing time slightly increases at the higher temperature, and homogenous optical linewidth decreases, which is also unusual for the majority of semiconductors. The insights generated by the reported simulations provide valuable contributions to our fundamental understanding of the remarkable properties that make hybrid perovskite extremely successful in solar energy applications. 20 ACS Paragon Plus Environment

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Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: XXXX Evolution of the VBM and CBM energy levels and overlap of the corresponding charge densities in MAPbI3 at 200 K and 300 K. The material is available free of charge via the Internet at http://pubs.acs.org.

Acknowledgements O. V. P. acknowledges funding from the U.S. Department of Energy, grant DESC0014429, and is grateful to the Donostia International Physics Center for support during a summer visit. J. F. T. acknowledges support by the National Nature Science Foundation of China, grant 51301066. D. C. acknowledges support from the Basque Government (project IT588-13) and IKERBASQUE, Basque Foundation for Science. The simulations were performed at the University of Southern California’s Center for High-Performance Computing (hpcc.usc.edu).



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