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Long Carrier Lifetimes in PbI-rich Perovskites Rationalized by Ab Initio Nonadiabatic Molecular Dynamics Chuan-Jia Tong, Linqiu Li, Limin Liu, and Oleg V. Prezhdo ACS Energy Lett., Just Accepted Manuscript • DOI: 10.1021/acsenergylett.8b00961 • Publication Date (Web): 09 Jul 2018 Downloaded from http://pubs.acs.org on July 9, 2018

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ACS Energy Letters

Long Carrier Lifetimes in PbI2-rich Perovskites Rationalized by Ab Initio Nonadiabatic Molecular Dynamics Chuan-Jia Tong1,3, Linqiu Li3, Li-Min Liu*1,2, Oleg V. Prezhdo*3 1

Beijing Computational Science Research Center, Beijing 100193, China School of Physics, Beihang University, Beijing 100191, China 3 Department of Chemistry, University of Southern California, Los Angeles, California 90089, United States 2

Email: [email protected], [email protected]

Abstract:

Hybrid organic-inorganic perovskites have attracted considerable interest due to

their impressive performance in solar energy applications. Many experiments show that a slight excess of PbI2 significantly enhances the properties of the most studied CH3NH3PbI3 compound. We use real-time time-dependent density functional theory and nonadiabatic molecular dynamics to demonstrate that the effect arises due to decreased electron-phonon interactions responsible for nonradiative charge recombination. The fast organic CH3NH3+ (MA) cations, present on surfaces of stochiometric and MAI-rich perovskites, are particularly mobile and introduce high frequency phonons and strong electric fields that couple to the charge carriers and create large nonadiabatic coupling. Excess PbI2 decreases MA surface coverage, reduces the nonadiabatic coupling by up to an order of magnitude, and extends the charge carrier lifetime. Generally, charges in perovskites are long-lived because the nonadiabatic coupling is very small, less than 1 meV, and quantum coherence formed during charge recombination is short, less than 10 fs. Our results rationalize why decreasing concentration of the organic cations on perovskite surfaces can suppress nonradiative charge carrier recombination and improve material performance.

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Methylammonium lead iodide CH3NH3PbI3 (MAPbI3) solar cells are attracting considerable interest because of their high photovoltaic performance.1-3 Following the first report of a perovskite solar cell in 2009 with the power conversion efficiency of 3.8%,4 it has a very quick development. After several key advances in the following years, the perovskite efficiency has increased rapidly now reaching beyond 22%.5 The state-of-the-art performance mainly stems from the large optical absorption cross-section,6-10 easy band-gap tuning by chemical composition,11-12 efficient charge-carrier generation,13-15 and very long electron and hole diffusion lengths.16-17 Apart from their excellent photo-electric properties, hybrid perovskites attract so much attention due to their low production cost.18-19 Although perovskite solar cells show an outstanding performance, there still remain several phenomena that require clear understanding, such as the current-voltage hysteresis20-25 and stability upon exposure to humidity.26-31 Recently, many experiments have reported beneficial effects of a slight excess of PbI2 in perovskite films.32-37 For example, Roldan-Carmona et al.32 observed an improvement of crystallinity and electron transfer by incorporating excess PbI2 as an additive in the perovskite film. Kim et al.36 showed an improved open-circuit voltage and field-factor by incorporating excess PbI2 into perovskite phase. They deduced that excess PbI2 can suppress nonradiative charge carrier recombination. However, the reason for why a few percent excess PbI2 is essential to achieve high efficiency devices remains unclear, motivating theoretical analysis of the microscopic mechanistic details of charge carrier recombination. In this letter, we report time-domain ab initio simulations of the nonradiative electron-hole recombination process in PbI2-rich and MAI-rich perovskites. The simulations show that the charge carrier lifetime is several times longer in the PbI2-rich system due to weaker nonadiabatic coupling (NAC). Surfaces of stoichiometric and MAI-rich systems contain significant concentrations of MA cations, which are mobile, exhibit high-frequency motions and create strong electric fields that couple to the charge carriers. Lower concentrations of MA cations in PbI2-rich systems decrease electron-phonon interactions and prolong charge carrier lifetimes. Other factors governing nonradiative relaxation, such as electronic energy gap and quantum coherence time, are less important in distinguishing the properties of the PbI2-rich and MAI-rich systems. The reported results rationalize the experimental observations at the atomistic level, and demonstrate how control over concentration of the organic cations on perovskite surfaces provides a mechanism to



improve performance of the perovskite materials. The electronic structure and molecular dynamics (MD) trajectories were obtained using plane-wave density functional theory (DFT) as implemented in the Quantum Espresso38 package. The exchange-correlation energy was calculated with the Perdew-Burke-Ernzerhof (PBE) functional39 and ultrasoft pseudopotentials. To improve description of the long-range van der Waals interactions, the DFT-D2 correction was employed.40 The periodic slab was constructed to represent the (001) surface of tetragonal MAPbI3. The slab was three layers thick and contained 2×2 MAPbI3 unit cells along the periodic x-y plane. A 20 Å vacuum layer was added in the z direction to eliminate interactions between slab images. The basis set energy cutoff was set to 50 Ry. A uniform 3×3×1 Monkhorst-Pack k-point mesh was used for the structural optimization, MD, and density of states (DOS) calculations. The NAC and NAMD calculations for the charge recombination dynamics were performed at the Γ-point only for computational efficiency, since MAPbI3 is a direct band-gap semiconductor with the conduction and valence band edges located at the Γ-point. Excitonic effects were not included into simulations both due to high computational cost and because exciton binding energy in hybrid perovskites is quite low, 20~60 meV41-43. This value is comparable to kBT at room temperature, and therefore, excitons in perovskites are either bound very weakly or exist as free electron-hole pairs. Analysis of the transient photoluminescence data has led to the conclusion that the properties of the excited states in lead halide perovskite are dominated by free charge pairs rather than bound excitons.44 The structures were fully relaxed until the calculated Hellmann-Feynman forces were smaller than 0.05 eV/Å. Then, the systems were heated to 300 K through repeated velocity rescaling. An adiabatic MD trajectory with a 1 fs atomic time-step was produced. The electron-hole recombination was modeled with the decoherence-induced surface hopping (DISH) approach45 to NAMD, implemented in the PYXAID package.46-47 1000 initial conditions were selected from the adiabatic MD trajectory, and NAMD simulations were performed using 300 random number sequences to sample surface hopping probabilities for each initial condition. The technique has been used successfully to study excited state dynamics in other perovskite systems.7, 17, 28, 48-54 We focus on the tetragonal phase of MAPbI3 at room temperature, as proposed in our previous report.55 According to the previous DFT calculations,56-57 two (001) surface models with different terminations are employed to represent PbI2-rich (Fig. 1(a)) and MAI-rich (Fig. 1(c))


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perovskites. Both models consist of three repeated layers of PbI6 octahedra with organic MA molecules between the layers. The surfaces of the MAI-rich perovskite are terminated with the MA molecules, while the surfaces of the PbI2-rich perovskite are terminated with the inorganic component. The partial densities of states (PDOS) of the PbI2-rich and MAI-rich perovskite structures are presented in Fig. 1(b) and 1(d). Consistent with the previous reports,58-59 only lead and iodine atoms contribute to the DOS around the valence band maximum (VBM) and the conduction band minimum (CBM). Essentially no contribution from the organic MA molecules are seen around the band-gap, suggesting that the MA molecules do not directly participate in the charge generation and recombination processes. Surfaces of perovskite do not introduce midgap states, as confirmed by our work and previous reports56-57, 60. In agreement with the previous results,60 the PbI2-rich system shows a smaller band-gap, which can be rationalized by regarding unsaturated chemical bonds of the surface Pb and I atoms as shallow trap states. One expects48 that the smaller band-gap should accelerate charge recombination in that PbI2-rich perovskites. However, many experiments suggest that the enhanced performance of PbI2-rich systems stems from slow nonradiative recombination and long lifetime of charge carriers. Therefore, the band-gap is not the main factor that determines the observed behavior, and it remains to be establish what other factors play key roles during charge carrier recombination in the perovskites.



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Figure 1. Side view of the optimized MAPbI3 structures with exposed (001) surfaces for two stoichiometric ratios: (a) high concentration of PbI2 (PbI2-rich), and (c) high concentration of MAI (MAI-rich). Partial density of states (PDOS) of (b) PbI2-rich and (d) MAI-rich perovskites. Lead is dark gray, iodine is purple, carbon is brown, nitrogen is blue, and hydrogen is pink.

In order to investigate in detail the nonradiative charge carrier recombination process, we performed NAMD simulations on both PbI2-rich and MAI-rich perovskite structures. The results are summarized in Table 1 and Fig. 2-4. The time-dependent population of the photo-excited state is shown in Fig. 2. The carrier recombination time τ is obtained using the short-time linear approximation to the exponential decay, f(t) = exp(-t/τ) ≈ 1 – t/τ. Compared with many other photocatalytic materials,61-63 both MAI-rich and PbI2-rich perovskites exhibit long, nanosecond recombination times, matching well the previous experimental42, 64-65 and theoretical7, 28, 48-50, 52, 66 reports. Further, we calculated the radiative lifetimes using the Einstein coefficient for spontaneous emission, and obtained 63.1 µs and 31.7 µs for the MAI-rich and PbI2-rich perovskites, respectively, in agreement with the previous experimental and theoretical reports.44, 67-69

The radiative lifetime is several orders of magnitude longer than the nonradiative lifetime,

indicating that nonradiative relaxation constitutes the main mechanism of charge recombination. Such long nonradiative and radiative recombination times result in large electron and hole diffusion lengths observed in the perovskite materials. The current calculations show that the excited state population decays by nearly an order of magnitude more slowly in the PbI2-rich than MAI-rich system, rationalizing the experimentally established enhanced performance of PbI2-rich perovskites.32-37

Table 1. Optimized geometry band-gap, and canonically averaged pure-dephasing time (τd), absolute value of NAC, and nonradiative electron-hole recombination time (τ) for the studied perovskites. form

band-gap (eV)

τd (fs)

|NAC| (meV)

τ (ns)

PbI2-rich

1.24

3.29

0.011

23.08

MAI-rich

1.78

3.57

0.096

3.56


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ACS Energy Letters

To elucidate the origin of the difference in the charge recombination rates in the PbI2-rich and MAI-rich systems, we consider in detail the factors affecting quantum transition rates. The rate of the nonradiative electron-hole recombination depends on electronic energy gap, quantum coherence, and NAC. Generally, loss of quantum coherence decreases transition rate.70-71 We estimate the quantum coherence time as the pure-dephasing time, τd, of the optical response theory by computing the 2nd order cumulant approximation to the pure-dephasing function,71 and fitting it by the Gaussian, D(t)=exp(-0.5(t/τd)2). The computed pure-dephasing functions are plotted in Fig. 3(a). The functions are nearly identical, and the fitted pure-dephasing times are very short, 3.29 fs in PbI2-rich and 3.57 fs in MAI-rich. The similar time scales indicate that a slight excess of either PbI2 or MAI has little effect on quantum coherence during electron-hole recombination in the perovskite. The pure-dephasing is very fast because the electrons and holes are localized on different parts of the inorganic subsystem, Pb and I respectively, and the overlap between their wavefunctions is small. Fast decoherence is an important factor in achieving the slow recombination rates and long charge carrier lifetimes in the perovskite materials.

Figure 2. Evolution of the excited state population in (a) PbI2-rich and (b) MAI-rich perovskites during the electron-hole recombination dynamics. The red dotted lines show linear fits.

Since there is essentially no difference in the quantum decoherence rates in the MAI-rich and PbI2-rich systems, and the lower band-gap in the PbI2-rich system favors faster relaxation, in contrast to the NAMD and experimental results, NAC must be the dominant factor rationalizing



the difference between the two cases. As shown in Table 1, the canonically averaged absolute NAC value is extremely small in the PbI2-rich perovskite, 0.011 meV. While also small, the corresponding value for the MAI-rich system is almost nine times larger, 0.096 meV. Thus indeed, the smaller NAC in the PbI2-rich perovskite explains its longer excited state lifetime and better photovoltaic performance. In addition to computing the averages, we monitor the NAC fluctuation during the dynamics. Fig. 3(b) presents the absolute NAC values over 3000 fs in both MAI-rich and PbI2-rich systems. The NAC fluctuates a lot, in particular, because it is proportional to nuclear velocity which changes significantly during MD. The difference between the instantaneous NAC values for the MAI-rich and PbI2-rich systems is very significant. The NAC in the PbI2-rich perovskite is always small, remaining less than 0.2 meV. In contrast, the NAC can exceed 1 meV in the MAI-rich perovskite, in particular, around 1600 fs and 2450 fs. In general, the small, sub-meV NAC values and rapid decoherence explain the long, nanosecond electron-hole recombination times in the perovskites.


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ACS Energy Letters

Figure 3. (a) Pure-dephasing functions. (b) Time-dependent absolute NAC. (c) Phonon influence spectra.

In order to characterize the nuclear motions that determine the large NAC values in the MAI-rich system, we checked several configurations during the 2200-2500 fs time interval when the NAC reaches high magnitude (see Fig. 3b). Fig. 4 demonstrates that the MA cations located on the perovskite surface exhibit the largest motions. Since the MA species are charged and carry significant dipole moments, they can couple strongly to electrons and holes. There exists a significant amount of theoretical and experimental work showing that the organic molecules can rotate inside inorganic cages at room temperature.72-74 Indeed, the MA molecules in both outmost



and subsurface layers rotate significantly, as shown by the curved arrows in Fig. 4. The MA molecules in the outmost layer exhibit larger amplitude motions and faster rotations, and even show a tendency for out-of-surface movement. For example, compared with the original configuration at 2200 fs, the outmost MA molecules show a significant displacement in the vertical direction during the next few hundreds of femtoseconds, as large as 1.38 Å at 2500 fs. The enhanced motions of the MA molecules on the perovskite surface induce strong electron-phonon interactions in the MAI-rich perovskites. Generally, charge separation and recombination are determined by both electronic coupling and electron-phonon interactions. The latter are particularly important for the recombination process. Therefore, we can conclude that it is the strong electron-phonon coupling induced by intense vibrations of the organic molecules that results in large NAC values in the MAI-rich perovskite, leading to faster electron-hole recombination.

Figure 4. Characteristic snapshots of the MAI-rich perovskite along the MD trajectory at successive times: (a) 2200 fs; (b) 2300 fs; (c) 2400 fs; (d) 2500 fs, in the region of large NAC (Fig. 3b). The dashed circles demonstrate the original arrangement of the MA molecules in (a). The curved and straight arrow indicate rotation and upward movement, respectively.

In order to confirm the above conclusions and to obtain further insights into the electron-phonon interactions in the two systems, we report the influence spectra calculated as Fourier transforms of the autocorrelation functions of the phonon-induced fluctuations of the electronic energy gaps. The influence spectra characterize the phonon modes that couple to the electronic subsystem. Presented in Fig. 3(c), the spectra exhibit dominant peaks near 100 cm-1 and 200 cm-1. Compared to the PbI2-rich system, the MAI-rich perovskite shows many signals in the


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ACS Energy Letters

high frequency area (> 250 cm-1). High-frequency phonons are particularly important, because they have higher velocities (for a given kinetic energy or temperature) and create larger NAC that is proportional to the velocity. The Pb-I inorganic sublattice of the MAPbI3 perovskite show only low frequency vibrations, with the Pb-I bending and stretching modes at 62 cm-1 and 90 cm-1, respectively.75 The higher frequency modes in the range between 250 and 400 cm-1 arise from the much lighter MA cations, in particular their hindered rotations.75-77 The modes above 500 cm-1 show very little intensity in the computed influence spectra. The broader range of phonon modes, and in particular, participation of the higher frequency modes, enhance electron-phonon interactions and increase NAC in the MAI-rich system. The excess of PbI2 used in the experiments decreases the concentration of MA molecules on perovskite surface, weakens electron-phonon interactions, and decreases the NAC responsible for the electron-hole recombination. This conclusion rationalizes the experimental findings.32-37 It is important to note that the effects of PbI2 excess on disorder and morphology of perovskite crystals are not trivial, and it is not absolutely clear if the PbI2 excess is incorporated in the perovskite upon its growth. The current work focuses on effect of a PbI2 excess on the relative stability and abundance of the differently terminated surfaces. Other processes related to formation of defects, both in the bulk and at the surface, upon the non-stoichiometric growth of the perovskite, can also play a prominent role in determining the properties of the final material. For example, a MAI deficient environment can hinder the formation of charge trap defects, such as iodine interstitials,78-79 leading to a concomitant improvement of the photoluminescence properties. In summary, using a combination of real-time TDDFT and NAMD, we have investigated the nonradiative electron-hole recombination in PbI2-rich and MAI-rich perovskites. The investigation has been motivated by multiple experimental publications which report longer charge carrier lifetimes and better photovoltaic performance of PbI2-rich perovskites. We have analyzed the three factors that govern the nonradiative charge carrier recombination, including the electronic energy gap, the nonadiabatic electron-phonon coupling, and quantum coherence. The analysis shows that the NAC is the main factor rationalizing the improved properties of the PbI2-rich systems. The coupling is nearly an order of magnitude smaller in the PbI2-rich perovskite compared to the MAI-rich perovskite. The electronic energy gap is smaller in the PbI2-rich system, having a



modest opposite effect on the charge carrier recombination, while quantum coherence is essentially the same in the two cases. Further analyses demonstrate that the increased NAC in the MAI-rich system can be attributed to stronger electron-phonon interactions induced by rotating MA cations, especially those exhibiting large amplitude motions on the perovskite surface. Accordingly, a slight excess of PbI2 in a perovskite decreases the concentration of MA molecules and removes them from the surface, weakening electron-phonon interactions. As a result, the decreased NAC suppresses nonradiative electron-hole recombination. In general, the charge carrier lifetimes in the perovskites are long because the NAC is very small, less than 1 meV, and quantum coherence during charge recombination is short, less than 10 fs. By providing a clear atomistic rationalization of the improved properties of the PbI2-rich perovskites, our work clarifies the experimental results and generates guidelines for improving the performance of the perovskite materials.

Acknowledgments C.-J. Tong and L.-M. Liu acknowledge support by the National Key Research and Development Program of China Grant No. 2016YFB0700700 and National Natural Science Foundation of China (No. 51572016 and U1530401). L.-Q. Li and O. V. Prezhdo acknowledge support of the U.S. National Science Foundation, Grant No. CHE-1565704. We acknowledge the computational support from the Tianhe-2JK computing time award at the Beijing Computational Science Research Center (CSRC) and the Special Program for Applied Research on Super Computation of the NSFC- Guangdong Joint Fund (the second phase). C.-J. Tong acknowledges financial support from the program (201604890017) of the China Scholarship Council that allowed him to visit the group of O. V. Prezhdo for one year.


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