Effects of ElectronâPhonon Coupling on Electronic Properties of

Subscriber access provided by YORK UNIV

Energy Conversion and Storage; Plasmonics and Optoelectronics

Effects of Electron-Phonon Coupling on Electronic Properties of Methylammonium Lead Iodide Perovskites Wissam A. Saidi, and Ali Kachmar J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b03164 • Publication Date (Web): 05 Dec 2018 Downloaded from http://pubs.acs.org on December 5, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry Letters

Effects of Electron-Phonon Coupling on Electronic Properties of Methylammonium Lead Iodide Perovskites Wissam A. Saidi*1 and Ali Kachmar2 1Department

of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States

2Qatar

Environment and Energy Research Institute (QEERI), Hamad Bin Khalifa

University (HBKU), Qatar Foundation, P.O. Box 5285, Doha, Qatar   * Correspondence to: [email protected]

ACS Paragon Plus Environment

1

The Journal of Physical Chemistry Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 17

Abstract Temperature can have a dramatic effect on the solar efficiency of methylammonium lead iodide (CH3NH3PbI3) absorbers due to changes in the electronic structure of the system even within the range of stability of a single phase. Herein using first principles density functional theory, we investigate the electron band structure of the tetragonal and orthorhombic phases of CH3NH3PbI3 as a function of temperature. The electron-phonon interactions are computed to all orders using a Monte Carlo approach, which is needed considering that the second-order Allen-Heine-Cardona theory in electron-phonon coupling is not adequate. Our results show that the band gap increases with temperature in excellent agreement with experimental results. We verified that anharmonic effects are only important near the tetragonal-cubic phase transition temperature. We also found that temperature has a significant effect on the effective masses and Rashba coupling. At room temperature, electron–phonon coupling is found to enhance the band effective mass by a factor of two, and to diminish the Rashba coupling by the same factor compared to T=0 K values. Our results underscore the significant impact of electron-phonon coupling on electronic properties of the hybrid perovskites. TOC Graphic


2

Page 3 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The organic-inorganic halide perovskites ABX3 such as methylammonium lead triiodide MAPbI3 (MA=CH3NH3) have an organic molecule A located at the center of a cube formed from corner sharing BX4 octahedra where B is a metal cation and X is a halide anion. These hybrid materials are promising solar absorbers with a current power conversion efficiency (PCE) exceeding 22.1%. 1-11 The high PCE of the hybrid perovskites, despite the relatively short period of active research on their photovoltaic properties, is due to their having many unique properties, which are favorable for solar applications such as high optical absorptions,12-13 small effective mass for charged carriers,14-15 long electronhole diffusion lengths,16-17 electronically benign grain boundaries,18-19 and shallow dominant point defects in crystalline 14, 20-21 and polycrystalline 19 systems. Additionally, the hybrid perovskites are susceptible to the Rashba effect 22-23 because their electronic band structure depends on spin-orbit interactions and they possess broken inversion symmetry. Previous studies showed that MAPbI3 has a large Rashba coupling of 2-3 eVÅ, which is attractive for applications in spintronic devices.24-27 Further, it was argued that Rashba splitting could also enhance the carrier lifetimes in MAPbI3 by 1 to 2 orders of magnitude.28-29 Temperature can modify the photovoltaic conversion efficiency due to changes in charge-carrier dynamics, absorption onset, and band alignment.30-32 Fundamental questions such as the nature of the electron–phonon coupling,32 electron–phonon scattering mechanism,33 and anharmonicity 34-36 are currently under scrutiny. These investigations are not only fundamental in nature, but also of practical importance considering that weather changes and ambient temperatures subject solar cells to a wide range of temperature changes from -20 to 40 °C. Further intrinsic losses in the solar conversion also increase the operating temperature of the solar cell. For example, the actual working temperature of a solar cell at room temperature with a PCE of 20 % exceeds 70 °C.37 In semiconductor solar cells, temperature increases have a negative effect on solar efficiencies.38 In MAPbI3 based solar cells, the open-circuit voltage (VOC) drops from 1.01 to 0.83 V as the temperature increases from 300 to 360 K.39 This was also confirmed in a recent study, which showed that VOC decreases linearly with temperature increase similar to conventional solar cells although in comparison the degradation is less in the perovskites.40 Therefore, an understanding of how temperature affects the photovoltaic properties of the hybrid perovskites is essential to understand the degradation mechanisms. The coupling between electrons and phonons leads to a renormalization of the electronic energy levels. Within the Born-Oppenheimer approximation, the electronic eigenenergy 𝜖𝒌𝑛 averaged over atomic vibrational states |𝜒𝒔⟩ at temperature 𝑇 can be written as: 1 ⟨𝜒 │𝜖 │𝜒 ⟩ 𝑒 ― 𝐸𝒔/𝑘𝐵𝑇, 𝜖𝒌𝑛(𝑇) = (1) ℤ 𝒔 𝒔 𝒌𝑛 𝒔 where ℤ=∑𝒔𝑒 ― 𝐸𝒔/𝑘𝐵𝑇 is the partition function and 𝑘𝐵 is the Boltzmann constant. This expression is typically approximated by an expansion in terms of the vibrational normal modes 𝑢𝒒𝜐. Truncating at second order in 𝑢𝒒𝜐 leads to the quadratic approximation, which can be efficiently evaluated using first-principles methods in conjunction with non-

∑


3


Page 4 of 17

diagonal supercells.41 Additionally, if the so-called rigid ion approximation is invoked, 𝜖𝒌𝑛 (𝑇) can be evaluated using density functional perturbation theory (DFPT); the resulting scheme is known as Allen-Heine-Cardona (AHC) theory.42, 43 For systems where the loworder expansion of Equation (1) is not sufficient to capture the temperature dependence of energy levels, it is still possible to evaluate Equation (1) directly using a Monte Carlo (MC) 1 𝑀 approach as 𝜖𝒌𝑛(𝑇) = 𝑀∑𝑖 𝜖𝒌𝑛 (𝑢𝑖), where 𝑀 is the number of the sampling points distributed according to the vibrational density of states.44 The MC approach is the most accurate approach, but this is at the expense of a larger computational cost especially to control finite size effects. Recently, two experimental studies showed that the band gap of MAPbI3 changes by 30-40 meV in the temperature range of solar cell operation.30, 31 Saidi, Poncé and Monserrat used first-principles density function theory (DFT) calculations to explain the experimental results, showing in particular that the commonly applied AHC theory significantly overestimates the band gap changes. This study showed that the failure of the AHC theory is not due to the rigid-ion approximation, spin-orbit coupling (SOC), or due to intrinsic errors in the exchange-correlation functional but is mainly due to the truncation of the electron–phonon expansion of Equation (1) to second order. Further, this study concluded that an excellent agreement with experiment for band gap renormalization is only obtained when including high-order terms in the electron–phonon interaction and spin-orbit coupling 32. While this study confirmed for the first time the importance of high-order terms in the electron–phonon coupling by direct comparison with experiment, the study was only applied for the cubic phase of MAPbI3, which is known to have imaginary phonon frequencies at T=0K reflecting its dynamical instability at low temperatures. This calls for more investigations on electron-phonon coupling in the stable low temperature MAPbI3 phases. Herein, we use first-principles density functional theory calculations in conjunction with van der Waals and SOC to study temperature effects on the electronic band structure of the low-temperature orthorhombic and tetragonal phases of MAPbI3. In addition to the band gap, we also study the impact of the electron–phonon coupling on Rashba coupling and carrier effective masses. We show that the MC approach can well describe the temperature renormalization effects on the band gap in the two phases yielding results in very good agreement with experimental results. Also, importantly, we show that thermal effects due to electron-phonon coupling are important for the effective masses as well as the Rashba splitting especially in the tetragonal phase compared to traditional semiconducting materials. This is expected to have significant impact on understanding charge transport properties in MAPbI3. Our density functional theory calculations are carried out using FHI-aims45-48 and CP2K.49 We employed the Perdew-Burke-Ernzerhof (PBE)50 functional plus dispersion corrections using Tkatchenko and Scheffler (TS) dispersion51-52 and Grimme DFT-D353 in FHI-aims and CP2K, respectively. Dispersion corrections have been previously shown to be important in the hybrid perovskites.54 We accounted for spin-orbit interactions as these are important to describe electron–phonon coupling.32 More details are provided in the supporting information.19, 32, 45-57


4

Page 5 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Results and Discussion. The PBE + SOC band structure of the orthorhombic and tetragonal MAPbI3 phases are shown in Figure 1. Both phases have a band gap at the Γ point of the Brillouin zone of magnitude 0.95 and 0.85 eV, respectively. The band gaps are underestimated compared to experimental results due to well-known problems of semilocal DFT functionals, which can be rectified for MAPbI3 to a large degree using hybrid functionals such as HSE or GW calculations.58 For reference, and to show the importance of the SOC corrections, the corresponding PBE band gaps of the two phases are 1.77 and 1.74 eV, receptively. The slightly larger band gap of the orthorhombic phase compared to the tetragonal one is consistent with previous calculations59 as well as with experimental results.31 From the projected density of states of the phases, we find that the conduction band minimum (CBM) has contributions from the p-band of Pb, while as valence band maximum (VBM) has contributions from the s-band of Pb and p-band of iodine, in agreement with several studies. While the band structures of the two phases are fairly similar, the orthorhombic phase is centrosymmetric and hence it has no Rashba splitting, as Figure 1 shows. The Rashba splitting has been shown to be sensitive to the crystal structure.27,60 The electronic properties of MAPbI3 are expected to vary with temperature due to thermal lattice expansion and electron–phonon coupling. Thermal expansion effects can be accounted for using quasi-harmonic approximation (QHA),62 by examining the band structure of the system as a function of the equilibrium lattice obtained at different temperatures. In the QHA, the Helmholtz free energy of a solid is the sum of the electronic energy and the vibration energy, and is thus a function of the lattice parameters and temperature. This dependence provides a potential energy surface that can be utilized to investigate the anharmonicity associated with the thermal expansion of the lattice due to the dependence of the harmonic frequencies on structural changes. Previously, Saidi and Choi investigated the temperature dependence of the lattice parameters of tetragonal and cubic phases of MAPbI3 using the QHA, and showed that the low-energy phonon modes due to the inorganic lattice are the main contributing factor for stabilizing the cubic phase at high temperatures.54 Using the temperature-dependent lattice parameters from this study, we examine the evolution of the electronic properties of MAPbI3 due to thermal expansion. Our results for the band gap, as well as for the effective masses and Rashba coupling show that lattice expansion leads to negligible effects on the band structure in comparison to the electron–phonon coupling. For example, using the equilibrium lattice parameters of the tetragonal phase (a; c) = (8:965; 12:601), (8.991; 12.640), and (9:000; 12:656) Å at T=0, 100 and 200 K, we find that the lattice expansion only increases the PBE + SOC band gap by 6 and 18 meV, which is an order of magnitude smaller than the corresponding changes 80 and 145 meV due to electron–phonon coupling. These results are also consistent with previous results on the cubic phase of MAPbI3.32 Additionally, we find that thermal expansion of the lattice has even a smaller effect on the effective masses of charge carriers and Rashba coupling. Therefore, given the relatively small effect of lattice expansion, we can safely apply the thermal expansion corrections obtained from the tetragonal phase to the orthorhombic phase. This approximation is quite


5


Page 6 of 17

useful given that the QHA investigations of the orthorhombic phase with three different lattice parameters is computationally not trivial. Figure 2 shows the temperature dependence of the band gap for the orthorhombic and tetragonal phases. The experimental results are from Ref. 43, which reported band gap measurements using transmittance and photoluminescence spectroscopy. The theoretical results are obtained using the MC scheme in conjunction with PBE and PBE + SOC by averaging over 20-60 different configurations sampled according to the vibrational density of states. We used more configurations at higher temperatures to obtain results with smaller statistical errors. To avoid well-known errors of PBE in computing band gaps, and to focus only on the band gap renormalization with temperature, we shift the calculated values such that the computed band gaps match the experimental results at 8K and at 160 K for the orthorhombic and the tetragonal phase, respectively. This is justified because we previously showed that PBE functional successfully describes temperature renormalization effects on the band structure of MAPbI3 yielding results comparable to those obtained at the HSE level.32 The band gap of MAPbI3 increases with temperature based on experimental studies, which is also nicely reproduced by our theoretical simulations, as Figure 2 shows. Previous results for the cubic phase showed that low-energy crystal modes dominate the coupling over the molecular high energy modes, which can be understood because the VBM and CBM are formed by states whose character is dominated by I and Pb, respectively. Between 8 and 140K, the bandgap of the orthorhombic phase increases by 53 and 67 meV based on transmittance and photoluminescence spectroscopy measurements resulting in a 0.45(5) meV/K linear rate change. Our calculations show changes of 49(7) and 57(9) meV at the PBE and PBE + SOC theory level resulting in a linear rate of change of 32(5) and 38(6) meV/K. For the tetragonal phase, both experiment and theory show that the band gap changes with temperature are slightly smaller than those in the orthorhombic phase. Experimentally, the band gap increases by 37 meV between 160 to 295 K with a linear band gap change rate of 0.27 meV/K. Our calculations show that between 150 and 250K, the band gap increases by 14(12) and 37(11) meV with PBE and PBE+SOC. Although the Monte Carlo approach of Equation (1) captures electron-phonon coupling to all orders, this is still only valid in the harmonic approximation. Previous studies have shown that anharmonicity is important for the hybrid perovskites 34-35. To check whether our findings are still valid by accounting for anharmonic region, we carried out long ab initio molecular dynamics (AIMD) simulations for 30 picosecond (ps) and monitored the band gap of the system. For the tetragonal phase, at T=300 K near the tetragonal-cubic phase transition, we carried out the AIMD for 100 ps to minimize statistical noise. As shown in Figure 2, the results obtained using the MD approach are in good agreement with the MC results for T=200 and 250 K. For T=300 K, the value obtained using AIMD is smaller than the value obtained using MC. This is likely because the tetragonal phase prematurely starts transforming into the cubic phase with the smaller band gap.31 This phase transition cannot be described using our MC approach, because it is still based on the harmonic approximation.


6

Page 7 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Using the temperature dependence of the energy levels as described in Equation (1), we also examined the variations with temperature of the band effective masses and Rashba splitting of the frontier orbitals. As mentioned before, temperature changes are mainly due to the electron–phonon coupling while as changes due to thermal expansion are negligible. We obtained the effective mass and Rashba splitting by fitting the split frontier bands to ℏ2 𝒌2/2 𝑚 ∗ + 𝛼𝑅|𝑘|.22, 63 The Rashba coupling 𝛼𝑅 can be expressed as 𝛼𝑅 = 2𝐸𝑅/𝑘𝑅 where 𝑘𝑅 is the distance in k-space between the crossing point of the spin-split conduction or valence bands, and 𝐸𝑅 is the corresponding energy difference as schematically shown in the inset of Figure 1 (b). Figure 3 shows the variations of the effective masses of the two MAPbI3 phases along high symmetry directions. Despite the presence of the organic molecule, we see that the effective masses retain the symmetry of the inorganic lattice and are nearly isotropic in the [110] plane for the tetragonal phase. The T=0 K values of the tetragonal or orthorhombic phases are slightly larger than those obtained at the static limit without the zero-point vibrations. Overall the results are in good agreement with previous investigations. 58-59, 61 Electron–phonon coupling enhances the band effective masses with temperature in the two phases. Further this increase is significant in the tetragonal phase resulting in effective masses at room temperature for either the electron or hole carriers larger by a factor of 2 compared to the T=0 K values. While the increase in the effective masses with temperature is consistent with other materials, 40, 60, 64-65 the changes in MAPbI3 are anomalous. For example, in silicon the effective mass increases by only ~2% between ~0 and 300 K.40, 60 Based on the Drude model, the increase in the effective mass with temperature is consistent with the decrease in the electron and hole mobility that was observed in the experimental results.31 However, a complete understanding of temperature dependence of the hole and electron mobilities requires also determining the temperature dependence of the momentum scattering time and how it decreases with temperature,66-67 which is beyond the current investigation. The temperature variations of the momentum shift 𝑘𝑅 and energy splitting 𝐸𝑅 due to Rashba effect are depicted in Figure 4. These are computed using PBE + SOC level, which was shown previously to render results in par with those of HSE functional.68 For the tetragonal phase, the static value 𝛼𝑅 for the hole/electron bands without the effects of electron–phonon coupling is 0.72/1.53 eVÅ in the [110] plane along Γ-Z or Γ-A, and 0.037/0.2 eV Å in the [001] plane along Γ-U (see Figure 1). Similar to the effective masses, the energy splitting 𝐸𝑅 and momentum shift 𝑘𝑅 in the tetragonal phase are isotropic in the [110] plane, and are significantly much larger than the corresponding values along the [001] axis. The reason for the smaller Rashba splitting from Γ(0,0,0) to U(0,0,1/2) is because the degeneracy of the bands in this direction is maintained, as can be seen from Figure 1. The static values of the Rashba splitting are consistent with previous studies 24-27 although there is a spread in the reported values considering that the splitting depends on the degree of the alignment of the organic cations.68 As seen from Figure 4, the Rashba splitting in the conduction bands are more pronounced than those of the valence bands due to the strong spin-orbit coupling in the conduction band with the dominant Pb character. 27, 58 Although the Rashba splitting of the orthorhombic phase is zero in the static limit, Figure


7


Page 8 of 17

4(a) and (b) show that electron-phonon coupling enhances the splitting because of the broken inversion symmetry. However, the band splittings are still appreciably small compared to the tetragonal phase. The induced Rashba splitting in the orthorhombic phase is similar in origin to the “dynamical spin splitting” in centrosymmetric systems such as CsPbCl3.69 The temperature dependence of 𝐸𝑅 and 𝑘𝑅 in Figure 4 is noticeable, both of which showing an increase with temperature. However, despite this increase, the Rashba coupling 𝛼𝑅 decreases with temperature because the rate of temperature increase in 𝑘𝑅 is more pronounced than that of 𝐸𝑅. This is consistent with the results for BiTeX (X=I, Br, Cl), which also showed that increasing temperature reduces the Rashba coupling.70 For the tetragonal phase, the Rashba coupling 𝛼𝑅 decreases by a linear rate of ~1.15(5) x10-3 eV Å/K for the two frontier orbitals resulting in a value at T=300 K that is 50% smaller than the static value. However, it is important to note that this analysis is valid in the harmonic approximation. Previous AIMD simulations showed that the dipoles will be randomly oriented with a rotational time scale of ~7 ps at 300 K.71 This would suggest that the Rashba coupling that is correlated with the long-range ferroelectric order of the system will also be negligible. In conclusion, we have determined the temperature dependence of the band gap, effective masses, and Rashba coupling of the orthorhombic and tetragonal phases of MAPbI3 using first-principles simulations. In addition to our previous results for the cubic phase of MAPbI3, we have shown that our approach based on Monte Carlo evaluation of Equation (1), which recovers electron–phonon coupling within the harmonic approximation to all orders, can successfully describe the increase of the band gap with temperature of all phases yielding results in excellent agreement with experimental studies. Further, we showed that we recover similar results for the band gap increase with temperature by including anharmonic effects from ab initio molecular dynamics simulations except at T=300 K near the phase transition temperature. Effects for thermal expansion are found to be less important in MAPbI3 for the band gaps, even for the high temperature cubic phase, as well as for effective masses of carriers and Rashba splitting. We also found that temperature have a strong effect on the effective masses and Rashba coupling. For the tetragonal phase, we showed that the room temperature effective mass increases by a factor of two, while as the Rashba coupling decrease by a factor of two compared to the T=0 K values. These studies highlight the importance of electron-phonon coupling on the hybrid perovskite especially for transport properties. Supporting Information Computational details, phonon band structure, structural analysis at finite temperature from AIMD simulations obtained using 2x2x2 and 3x3x3 supercells. Acknowledgements WAS acknowledges many useful discussions with Dr. B. Monserrat, and thanks Dr. L. Herz for sending the data for Ref. 17. WAS also acknowledges a start-up fund from the Department of Mechanical Engineering and Materials Science at the University of Pittsburgh. We are grateful for computing time provided in part by the Extreme Science


8

Page 9 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation (#NSF OCI-1053575), Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC0206CH11357. AK acknowledges the HPC resources and services used in this work, which were provided by the Research Computing group in Texas A&M University at Qatar. Research computing is funded by the Qatar Foundation for Education, Science, and Community Development. Declaring Financial Interests The authors declare no competing financial interests. Corresponding Authors Wissam A. Saidi ([email protected]) References (1). Kojima, A.; Teshima, K.; Shirai, Y.; Miyasaka, T. Organometal halide perovskites as visible-light sensitizers for photovoltaic cells. J. Am. Chem. Soc. 2009, 131 (17), 60506051. (2). Im, J.-H.; Lee, C.-R.; Lee, J.-W.; Park, S.-W.; Park, N.-G. 6.5% efficient perovskite quantum-dot-sensitized solar cell. Nanoscale. 2011, 3 (10), 4088-4093. (3). Lee, M. M.; Teuscher, J.; Miyasaka, T.; Murakami, T. N.; Snaith, H. J. Efficient hybrid solar cells based on meso-superstructured organometal halide perovskites. Science. 2012, 338 (6107), 643-647. (4). Kim, H.-S.; Lee, C.-R.; Im, J.-H.; Lee, K.-B.; Moehl, T.; Marchioro, A.; Moon, S.J.; Humphry-Baker, R.; Yum, J.-H.; Moser, J. E.; et al. Lead iodide perovskite sensitized all-solid-state submicron thin film mesoscopic solar cell with efficiency exceeding 9%. Sci. Rep. 2012, 2, 591. (5). Etgar, L.; Gao, P.; Xue, Z.; Peng, Q.; Chandiran, A. K.; Liu, B.; Nazeeruddin, Md. K.; Grätzel, M. Mesoscopic CH3NH3PbI3/TiO2 heterojunction solar cells. J. Am. Chem. Soc. 2012, 134 (42), 17396-17399. (6). Ball, J. M.; Lee, M. M.; Hey, A.; Snaith, H. J. Low-temperature processed mesosuperstructured to thin-film perovskite solar cells. Energy & Environ. Sci. 2013, 6 (6), 1739-1743. (7). Heo, J. H.; Im, S. H.; Noh, J. H.; Mandal, T. N.; Lim, C.-S.; Chang, J. A.; Lee, Y. H.; Kim, H.-j.; Sarkar, A.; Nazeeruddin, Md. K.; et al. Efficient inorganic-organic hybrid heterojunction solar cells containing perovskite compound and polymeric hole conductors. Nat. Photon. 2013, 7 (6), 486-491. (8). Liu, M.; Johnston, M. B.; Snaith, H. J. Efficient planar heterojunction perovskite solar cells by vapour deposition. Nature. 2013, 501 (7467), 395-398. (9). Burschka, J.; Pellet, N.; Moon, S. J.; Humphry-Baker, R.; Gao, P.; Nazeeruddin, M. K.; Gratzel, M. Sequential deposition as a route to high-performance perovskitesensitized solar cells. Nature. 2013, 499 (7458), 316-319.


9


Page 10 of 17

(10). Haruyama, J.; Sodeyama, K.; Han, L.; Tateyama, Y. First-principles study of ion diffusion in perovskite solar cell sensitizers. J. Am. Chem. Soc. 2015, 137 (32), 1004810051. (11). Li, X.; Bi, D.; Yi, C.; Décoppet, J.-D.; Luo, J.; Zakeeruddin, S. M.; Hagfeldt, A.; Grätzel, M. A vacuum flash–assisted solution process for high-efficiency large-area perovskite solar cells. Science. 2016, 353 (6294), 58-62. (12). Yin, W. J.; Shi, T.; Yan, Y. Unique properties of halide perovskites as possible origins of the superior solar cell performance. Adv. Mater. 2014, 26 (27), 4653-4658. (13). Grancini, G.; Marras, S.; Prato, M.; Giannini, C.; Quarti, C.; De Angelis, F.; De Bastiani, M.; Eperon, G. E.; Snaith, H. J.; Manna, L.; et al.The impact of the crystallization processes on the structural and optical properties of hybrid perovskite films for photovoltaics. J. Phys. Chem. Lett. 2014, 5 (21), 3836-3842. (14). Du, M. H. Efficient carrier transport in halide perovskites: theoretical perspectives. J. Mater. Chem. A. 2014, 2 (24), 9091-9098. (15). Giorgi, G.; Fujisawa, J.-I.; Segawa, H.; Yamashita, K. Small photocarrier effective masses featuring ambipolar transport in methylammonium lead iodide perovskite: A density functional analysis. J. Phys. Chem. Lett. 2013, 4 (24), 4213-4216. (16). Xing, G.; Mathews, N.; Sun, S.; Lim, S. S.; Lam, Y. M.; Grätzel, M.; Mhaisalkar, S.; Sum, T. C. Long-range balanced electron-and hole-transport lengths in organicinorganic CH3NH3PbI3. Science. 2013, 342 (6156), 344-347. (17). Stranks, S. D.; Eperon, G. E.; Grancini, G.; Menelaou, C.; Alcocer, M. J.; Leijtens, T.; Herz, L. M.; Petrozza, A.; Snaith, H. J. Electron-hole diffusion lengths exceeding 1 micrometer in an organometal trihalide perovskite absorber. Science. 2013, 342 (6156), 341-344. (18). Guo, Y.; Wang, Q.; Saidi, W. A. Structural stabilities and electronic properties of high-angle grain boundaries in perovskite cesium lead halides. J. Phys. Chem. C. 2017, 121 (3), 1715-1722. (19). Shan, W.; Saidi, W. A. Segregation of native defects to the grain boundaries in methylammonium lead iodide perovskite. J. Phys. Chem. Lett. 2017, 8 (23), 5935-5942. (20). Yin, W.-J.; Shi, T.; Yan, Y. Unusual defect physics in CH3NH3PbI3 perovskite solar cell absorber. Appl. Phys. Lett. 2014, 104 (6), 063903. (21). Kim, J.; Lee, S. H.; Lee, J. H.; Hong, K. H. The role of intrinsic defects in methylammonium lead iodide perovskite. J. Phys. Chem. Lett. 2014, 5 (8), 1312-1317. (22). Rashba, E. Properties of semiconductors with an extremum loop. 1. Cyclotron and combinational resonance in a magnetic field perpendicular to the plane of the loop. Sov. Phys. Solid State. 1960, 2, 1109-1122. (23). Dresselhaus, G. Spin-orbit coupling effects in zinc blende structures. Phys. Rev. 1955, 100 (2), 580-586. (24). Frohna, K.; Deshpande, T.; Harter, J.; Peng, W.; Barker, B. A.; Neaton, J. B.; Louie, S. G.; Bakr, O. M.; Hsieh, D.; Bernardi, M. Inversion symmetry and bulk Rashba effect in methylammonium lead iodide perovskite single crystals. Nat. Commun. 2018, 9 (1), 1829. (25). Kim, M.; Im, J.; Freeman, A. J.; Ihm, J.; Jin, H. Switchable S = 1/2 and J = 1/2 Rashba bands in ferroelectric halide perovskites. PNAS. 2014, 111 (19), 6900-6904. (26). Kepenekian, M.; Robles, R.; Katan, C.; Sapori, D.; Pedesseau, L.; Even, J. Rashba and Dresselhaus effects in hybrid organic–inorganic perovskites: from basics to devices. ACS Nano. 2015, 9 (12), 11557-11567.


10

Page 11 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(27). Amat, A.; Mosconi, E.; Ronca, E.; Quarti, C.; Umari, P.; Nazeeruddin, M. K.; Grätzel, M.; De Angelis, F. Cation-induced band-gap tuning in organohalide perovskites: Interplay of spin–orbit coupling and octahedra tilting. Nano Lett. 2014, 14 (6), 3608-3616. (28). Zheng, F.; Tan, L. Z.; Liu, S.; Rappe, A. M. Rashba spin–orbit coupling enhanced carrier lifetime in CH3NH3PbI3. Nano Lett. 2015. (29). Azarhoosh, P.; McKechnie, S.; Frost, J. M.; Walsh, A.; van Schilfgaarde, M. Research update: Relativistic origin of slow electron-hole recombination in hybrid halide perovskite solar cells. APL Materials 2016, 4 (9), 091501. (30). Foley, B. J.; Marlowe, D. L.; Sun, K.; Saidi, W. A.; Scudiero, L.; Gupta, M. C.; Choi, J. J. Temperature dependent energy levels of methylammonium lead iodide perovskite. Appl. Phys. Lett. 2015, 106 (24), 243904. (31). Milot, R. L.; Eperon, G. E.; Snaith, H. J.; Johnston, M. B.; Herz, L. M. Temperature-dependent charge-carrier dynamics in CH3NH3PbI3 Perovskite thin films. Adv. Func. Mater. 2015, 25 (39), 6218-6227. (32). Saidi, W. A.; Poncé, S.; Monserrat, B. Temperature dependence of the energy levels of methylammonium lead iodide perovskite from first-principles. J. Phys. Chem. Lett. 2016, 7 (24), 5247-5252. (33). Brenner, T. M.; Egger, D. A.; Rappe, A. M.; Kronik, L.; Hodes, G.; Cahen, D. Are mobilities in hybrid organic–inorganic halide perovskites actually “high”? J. Phys. Chem. Lett. 2015, 6, 4754-4757. (34). Sendner, M.; Nayak, P. K.; Egger, D. A.; Beck, S.; Muller, C.; Epding, B.; Kowalsky, W.; Kronik, L.; Snaith, H. J.; Pucci, A.; et al. Optical phonons in methylammonium lead halide perovskites and implications for charge transport. Materials Horizons. 2016, 3 (6), 613-620. (35). Ivanovska, T.; Quarti, C.; Grancini, G.; Petrozza, A.; De Angelis, F.; Milani, A.; Ruani, G. Vibrational response of methylammonium lead iodide: from cation dynamics to phonon-phonon interactions. ChemSusChem. 2016, 9 (20), 2994-3004. (36). Carignano, M. A.; Aravindh, S. A.; Roqan, I. S.; Even, J.; Katan, C. Critical fluctuations and anharmonicity in lead iodide perovskites from molecular dynamics supercell simulations. J. Phys. Chem. C. 2017, 121 (38), 20729-20738. (37). Skoplaki, E.; Boudouvis, A. G.; Palyvos, J. A. A simple correlation for the operating temperature of photovoltaic modules of arbitrary mounting. Sol. Energy Mater. Sol. Cells. 2008, 92 (11), 1393-1402. (38). Nelson, J. The physics of solar cells; Imperial College Press: London, U.K.; 2003, 1-384. (39). Zhang, H.; Qiao, X.; Shen, Y.; Moehl, T.; Zakeeruddin, S. M.; Gratzel, M.; Wang, M. Photovoltaic behaviour of lead methylammonium triiodide perovskite solar cells down to 80 K. J. Mater. Chem. A. 2015, 3, 11762-11767. (40). Schwenzer, J. A.; Rakocevic, L.; Gehlhaar, R.; Abzieher, T.; Gharibzadeh, S.; Moghadamzadeh, S.; Quintilla, A.; Richards, B. S.; Lemmer, U.; Paetzold, U. W. Temperature variation-induced performance decline of perovskite solar cells. ACS Appl. Mater. Interfaces. 2018, 10 (19), 16390-16399. (41). Lloyd-Williams, J. H.; Monserrat, B. Lattice dynamics and electron-phonon coupling calculations using nondiagonal supercells. Phys. Rev. B. 2015, 92 (18), 184301. (42). Allen, P. B.; Heine, V. Theory of the temperature dependence of electronic band structures. J. Phys. C: Solid State Physics. 1976, 9 (12), 2305-2312.


11


Page 12 of 17

(43). Allen, P. B.; Cardona, M. Theory of the temperature dependence of the direct gap of germanium. Phys. Rev. B. 1981, 23 (4), 1495-1505. (44). Monserrat, B.; Engel, E. A.; Needs, R. J. Giant electron-phonon interactions in molecular crystals and the importance of nonquadratic coupling. Phys. Rev. B. 2015, 92 (14), 140302. (45). Blum, V.; Gehrke, R.; Hanke, F.; Havu, P.; Havu, V.; Ren, X.; Reuter, K.; Scheffler, M. Ab initio molecular simulations with numeric atom-centered orbitals. Comput. Phys. Comm. 2009, 180 (11), 2175-2196. (46). Havu, V.; Blum, V.; Havu, P.; Scheffler, M. Efficient O(N) integration for allelectron electronic structure calculation using numeric basis functions. J. Comput. Phys. 2009, 228 (22), 8367-8379. (47). Ren, X.; Rinke, P.; Blum, V.; Wieferink, J.; Tkatchenko, A.; Sanfilippo, A.; Reuter, K.; Scheffler, M. Resolution-of-identity approach to Hartree–Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions. New J. Phys. 2012, 14 (5), 053020. (48). Marek, A.; Blum, V.; Johanni, R.; Havu, V.; Lang, B.; Auckenthaler, T.; Heinecke, A.; Bungartz, H.-J.; Lederer, H. The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science. Journal of Physics: Condensed Matter. 2014, 26 (21), 213201. (49). VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J. Quickstep: fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comp. Phys. Comm. 2005, 167, 103-128. (50). Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865-3868 (51). Tkatchenko, A.; Scheffler, M. Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data. Phys. Rev. Lett. 2009, 102 (7), 073005. (52). Al-Saidi, W.; Voora, V. K.; Jordan, K. D. An assessment of the vdW-TS method for extended systems. J. Chem. Theory Comput. 2012, 8 (4), 1503-1513. (53). Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 2010, 132 (15), 154104. (54). Saidi, W. A.; Choi, J. J. Nature of the cubic to tetragonal phase transition in methylammonium lead iodide perovskite. J. Chem. Phys. 2016, 145, 144702. (55). VandeVondele, J.; Hutter, J. Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases. J. Chem. Phys. 2007, 127, 114105. (56). Goedecker, S.; Teter, M.; Hutter, J. Separable dual-space gaussian pseudopotentials. Phys. Rev. B. 1996, 54 (3), 1703-1710. (57). Carignano, M. A.; Kachmar, A.; Hutter, J. Thermal effects on CH3NH3PbI3 perovskite from ab initio molecular dynamics simulations. J. Phys. Chem. C. 2015, 119, 8991. (58). Umari, P.; Mosconi, E.; De Angelis, F. Relativistic GW calculations on CH3NH3PbI3 and CH3NH3SnI3 perovskites for solar cell applications. Sci. Rep. 2014, 4, 4467.


12

Page 13 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(59). Geng, W.; Zhang, L.; Zhang, Y.-N.; Lau, W.-M.; Liu, L.-M. First-principles study of lead iodide perovskite tetragonal and orthorhombic phases for photovoltaics. J. Phys. Chem. C. 2014, 118 (34), 19565-19571. (60). Quarti, C.; Mosconi, E.; De Angelis, F. Interplay of orientational order and electronic structure in methylammonium lead iodide: implications for solar cell operation. Chemistry of Materials. 2014, 26 (22), 6557-6569. (61). Menéndez-Proupin, E.; Palacios, P.; Wahnón, P.; Conesa, J. C. Self-consistent relativistic band structure of the CH3NH3PbI3 perovskite. Physical Review B 2014, 90 (4), 045207. (62). Allen, R. E.; De Wette, F. Calculation of dynamical surface properties of noble-gas crystals. I. The quasiharmonic approximation. Phys. Rev. 1969, 179 (3), 873. (63). Casella, R. C. Toroidal energy surfaces in crystals with wurtzite symmetry. Phys. Rev. Lett. 1960, 5 (8), 371-373. (64). Hofmann, T.; Kühne, P.; Schöche, S.; Chen, J.-T.; Forsberg, U.; Janzén, E.; Ben Sedrine, N.; Herzinger, C. M.; Woollam, J. A.; Schubert, M.; Darakchieva, V. Temperature dependent effective mass in AlGaN/GaN high electron mobility transistor structures. Appl. Phys. Lett. 2012, 101 (19), 192102. (65). Green, M. A. Intrinsic concentration, effective densities of states, and effective mass in silicon. J. Appl. Phys. 1990, 67 (6), 2944-2954. (66). Yu, Z.-G. Rashba effect and carrier mobility in hybrid organic–inorganic perovskites. J. Phys. Chem. Lett. 2016, 7 (16), 3078-3083. (67). Herz, L. M. Charge-carrier mobilities in metal halide perovskites: fundamental mechanisms and limits. ACS Energy Letters. 2017, 2 (7), 1539-1548. (68). Leppert, L.; Reyes-Lillo, S. E.; Neaton, J. B. Electric field- and strain-induced rashba effect in hybrid halide perovskite. J. Phys. Chem. Lett. 2016, 7 (18), 3683-3689. (69). Monserrat, B.; Vanderbilt, D. Phonon-assisted spin splitting in centrosymmetric crystals. arXiv:1711.06274, 2018. (70). Monserrat, B.; Vanderbilt, D. Temperature dependence of the bulk Rashba splitting in the bismuth tellurohalides. Phys. Rev. Materials. 1, 2017, 054201. (71). Govinda, S.; Kore, B. P.; Bokdam, M.; Mahale, P.; Kumar, A.; Pal, S.; Bhattacharyya, B.; Lahnsteiner, J.; Kresse, G.; Franchini, C.; Pandey, A.; Sarma, D. D. Behavior of methylammonium dipoles in MAPbX3 (X = Br and I). J. Phys. Chem. Lett. 2017, 8 (17), 4113-4121.


13


Page 14 of 17

Figures

Figure 1. Band structure of the orthorhombic and tetragonal phases of MAPbI using PBE+SOC. The 3

high symmetry points are defined as A=(1/2,0,0), Γ=(0,0,0), Z=(0,1/2,0), Q=(0,1/2,1/2), X=(1/2,1/2,1/2) and U=(0,0,1/2).


14

Page 15 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2. Temperature dependence of the band gap of orthonormal and tetragonal MAPbI3. The experimental results are from Milot et al. obtained using transmittance (T) and photoluminescence (PL) spectroscopy.30 The theoretical results are obtained using the MC approach based on PBE and PBE + SOC, as well as based on AIMD/PBE trajectory. Dispersion corrections are included in all calculations as described in the text. The theoretical results are shifted such that the experimental and theoretical results agree at T=8K and T=160K for two phases. The experimental results are connected by a line to guide the eye. The statistical error bars are included in all computed data points, but the size of some is smaller than symbol size. symbol size for some. J. Phys. Chem. Lett. Letter DOI


15


Page 16 of 17

Figure 3. Temperature dependence of the effective masses at the Gamma point evaluated along three high symmetry directions Γ to A=(1/2,0,0), Γ to Z=(0,1/2,0) and Γ to U=(0,0,1/2) using PBE + SOC for (a) orthorhombic and (b) tetragonal MAPbI3 phase. The values are averaged over the two-split valence (positive, hole) and conduction bands (negative, electron).


16

Page 17 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 4. Rashba energy splitting (a, c), and momentum splitting (b, d) for the orthorhombic (a, b) and tetragonal MAPbI3 phases. The blue bars are for values along [100], red bars along [010], and green bars along [001] directions. The momentum splitting is multiplied by 1000 for convenience. The legend for all subfigures is shown in (a).


17

Effects of ElectronâPhonon Coupling on Electronic Properties of

Recommend Documents

Effects of ElectronâPhonon Coupling on Electronic Properties of