Rapid Decoherence Suppresses Charge Recombination in Multi

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Rapid Decoherence Suppresses Charge Recombination in MultiLayer 2D Halide Perovskites: Time-Domain Ab Initio Analysis Zhaosheng Zhang, Weihai Fang, Marina Tokina, Run Long, and Oleg V. Prezhdo Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b00035 • Publication Date (Web): 13 Mar 2018 Downloaded from http://pubs.acs.org on March 13, 2018

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Nano Letters

Rapid Decoherence Suppresses Charge Recombination in Multi-Layer 2D Halide Perovskites: Time-Domain Ab Initio Analysis

Zhaosheng Zhang,1 Wei-Hai Fang,1 Marina V. Tokina,2 Run Long1∗, Oleg V. Prezhdo2

1

College of Chemistry, Key Laboratory of Theoretical & Computational Photochemistry of Ministry of Education, Beijing Normal University, Beijing, 100875, P. R. China

2

∗

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

Corresponding author, E-mail: [email protected]; Phone: +86-15712907817

1

ACS Paragon Plus Environment

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ABSTRACT: Two-dimensional (2D) Ruddlesden-Popper halide perovskites are appealing candidates for optoelectronics and photovoltaics. Nonradiative electron-hole recombination constitutes a major pathway for charge and energy losses in these materials. Surprisingly, experimental recombination is slower in multi-layers than mono-layer, even though multi-layer systems have smaller energy gaps and higher frequency phonons that should accelerate the recombination. Focusing on (BA)2(MA)n–1PbnI3n+1 with n=1 and 3, BA=CH3(CH2)3NH3 and MA=CH3NH3, we show that it is the enhancement of elastic electron-phonon scattering that suppresses charge recombination for n=3, by causing rapid loss of electronic coherence. The scattering is enhanced in the multi-layer 2D perovskites because, in contrast to the mono-layer, they contain MA cations embedded into the inorganic Pb-I lattice. Although MAs do not contribute directly to electron and hole wavefunctions, they perturb the Pb-I lattice and create strong electric fields that interact with the charges. The rapid loss of coherence explains long excited state lifetimes that extend into nanoseconds. Both electron-hole recombination and coherence times show excellent agreement with the corresponding lifetime and linewidth measurements. The simulations rationalize the observed dependence of excited state lifetime in 2D layered halide perovskites on layer thickness and advance our understanding of the atomistic mechanisms underlying charge-phonon dynamics in nanoscale materials. Keywords: 2D layered halide perovskites; quantum coherence; electron-hole recombination; elastic and inelastic electron-phonon scattering; non-adiabatic molecular dynamics; timedependent density functional theory 2


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Hybrid organic-inorganic halide perovskite solar cells have received great attention due to high power conversion efficiencies, growing rapidly to 22.1%1-9 within just six years since the first report of methyl ammonium halide sensitized cell in 2009.10 The rapid growth stems from utilization of the advantageous properties of perovskites,3, 11-14 including strong light absorption,3, 11

long carrier diffusion,12, 13 and slow electron-hole recombination.14 The easy, inexpensive and

low temperature solution-processing technology5, 6, 15 allows one to fabricate perovskite solar cells on a large scale. Three-dimensional (3D) organic-inorganic halide perovskites suffer from poor stability against humidity15, 16 and photo-irradiation.17 They are toxic due to presence of lead atoms.18,

19

Two-dimensional (2D) Ruddlesden-Popper perovskites have emerged as an

alternative to the 3D perovskites. The 2D systems exhibit the majority of the key photovoltaic properties of the 3D perovskites, while at the same time, they are more stable and less toxic due to reduced lead content.20, 21 2D perovskites are a class of layered materials with the chemical formula of AnA’n-1MnX3n+1 (n=1, 2, …∞),22, 23 in which A and A’ are organic cations, M is metal and X is halide. The value of n characterizes the number of the inorganic layers sandwiched between the long organic chains A. Such structures can be viewed as multiple quantum wells that produce stable excitons at room temperature.24, 25 The number of A’ cations is not necessarily one less than the number of A cations. If the number of A cations is fixed to 2 in a typical 2D perovskite BA2MAn-1PbnI3n+1,25 variation of n leads to monolayer BA2PbI4 (n=1), bilayer BA2MAPb2I7 (n=2), and tri-layer BA2MA2Pb3I10 (n=3). Compared to the 3D perovskites, the 2D perovskites offer additional flexibility in controlling the physical and chemical properties. For 3


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instance, the layer thickness affects dielectric environment and the optical bandgap.26, 27 Many efforts have been devoted to synthesize different 2D Ruddlesden-Popper perovskites.20, 21, 28 Among dozens of 2D perovskites,17, 20, 21, 25, 26, 29-32 (BA)2(MA)n–1PbnI3n+1, where BA=CH3(CH2)3NH3 and MA=CH3NH3, has received particular attention.31 Cao et al.21 have synthesized a series of layered (BA)2(MA)n–1PbnI3n+1 (n = 1, 2, 3 and 4) materials that exhibit strong light absorption under irradiation by visible light, achieving a 4.02% solar power conversion efficiency with the n = 3 compound. The layered structure of the 2D perovskites affords self-assembly perpendicular to a substrate, leading to ultrahigh surface coverage and improved light-harvesting.21 Blancon et al.26 have increased the conversion efficiency to 10% using (BA)2(MA)n−1PbnI3n+1 with n = 5. The increased thickness enhanced charge conductivity and reduced the optical bandgap, allowing the materials to harvest light in the near-IR region. Stoumpos et al.32 have determined the crystal structures of (BA)2(MA3)n−1PbnI3n+1 (n = 2, 3, and 4) using single crystal X-ray diffraction and calculated ab initio band structure. Shortly after, Lin et al.17 have demonstratted that (BA)2(MA)3Pb4I13 remains highly stable at elevated temperatures, in particular, compared to films of the commonly used CH3NH3PbI3 3D perovskite, due to absence of significant ion transport and presence of long hydrophobic BA cation chains, which improve moisture resistance of the 2D perovskite membrane and inhibit entry of oxygen. Using time-resolved photoluminescence spectroscopy, Guo et al.25 studied the influence

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of electron-phonon scattering on luminescence decay in layered (BA)2(MA)n−1PbnI3n+1 (n = 1, 2 and 3). They showed that the electron-hole recombination rate depends on the 2D perovskite thickness, with the excited-state lifetime ranging from 250 ps with n=1 to nanoseconds with n=3 at room temperature. The longer lifetime of the three-layer 2D perovskite compared to the monolayer is unexpected for several reasons. First, the electronic energy gap between the excited and ground states is smaller for n=3, suggesting faster relaxation, particularly since gap is large and the system is in the inverted Marcus regime.33 Most solid state and nanoscale materials with sufficiently large bandgaps are in the inverted regime, in which relaxation occurs by a nonadiabatic transition that is analogous to tunneling in the Marcus parabolas representation. The system never reaches a transition state defined by the parabola crossing, and the electronic energy is deposited into multiple vibrational modes. Second, the inorganic Pb-I lattice is more extended for n=3 than n=1, supporting a wider range of phonon modes that can promote the nonradiative charge recombination. Third, (BA)2(MA)2Pb3I10 contains light organic MA cations embedded into the inorganic lattice, while (BA)2PbI4 contains no such cations. The MA cations should accelerate the relaxation further, because they couple significantly to electrons and holes by generating strong electric fields, even though they do not contribute to the valence and conduction band edges.34-39 Nevertheless, the electron-hole recombination is much slower in (BA)2(MA)2Pb3I10 than (BA)2PbI4,25 requiring an explanation. Elucidating the mechanisms controlling charge recombination in the 2D perovskites is particularly important, since the recombination limits material quality in optoelectronics and light-to-current conversion 5


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efficiency in solar energy applications. In this letter, we report an ab initio time-domain study of non-radiative electron-hole recombination in mono- and tri-layer 2D perovskites, directly mimicking the experiment.25 We demonstrate that the tri-layer system has a smaller bandgap and a stronger charge-phonon coupling due to participation of a broader range of phonon modes. Nevertheless, the charge recombination is slower in the tri-layer, as in the experiment,25 due to faster loss of quantum coherence between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) . The fast motions of the MA cations embedded into the Pb-I lattice facilitate rapid elastic charge-phonon scattering that induces fast coherence loss in (BA)2(MA)2Pb3I10. Such motions are not available in (BA)2PbI4, which exhibits longer coherence and faster charge recombination. Additional higher frequency phonons appear in the Pb-I lattice of the tri-layer because it is stiffer and thicker than the mono-layer. The calculated coherence times are directly related to the homogeneous linewidths, showing excellent agreement with the experiment.25 We predict that the heterogeneous contributions to the linewidths are similar to the homogeneous contributions. To simulate electron-hole recombination in (BA)n(MA)n-1PbnI3n+1 (n = 1, 3) we employ decoherence induced surface hopping (DISH)40 implemented within the time-dependent KohnSham theory.41 The lighter and faster electrons are described quantum mechanically, and the heavier and slower nuclei are treated semi-classically.42-46 DISH incorporates quantum

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decoherence of the electronic subsystem due to interaction with nuclei, and interprets it as the source of nuclear trajectory branching resulting in surface hops.40, 47 Decoherence is required, because it is significantly faster than the non-radiative quantum transition taking place across a large energy gap.48, 49 The decoherence timescale is estimated as the pure-dephasing time of the optical response theory.50 The inverse of the pure-dephasing time gives the homogeneous optical linewidth that can be obtained experimentally.51, 52 DISH40 is implemented45 within the classical path approximation, which greatly reduces the computational effort, allowing one to using a precalculated ground state trajectory to drive the non-equilibrium electron-vibrational dynamics. The approximation is justified by the fact that thermal fluctuations in atomic coordinates exceed the difference in the system geometries associated with the ground and excited electronic states.44 The approach has been applied to study photoexcitation dynamics in a broad range of systems,48, 53-64 including CH3NH3PbI3 perovskites interfaced with TiO2,37, 39 containing grain boundaries,34 dopants,36 and defects,38 in contact with water molecules,34 exhibiting ordered phases,35 and forming polarons.38 Geometry optimization, adiabatic molecular dynamics (MD) and non-adiabatic (NA) couplings were obtained using the Vienna ab initio simulation package (VASP).65 The exchangecorrelation interactions were represented by the Pedrew-Burke-Ernzerh (PBE) functional.66 The projector-augmented wave (PAW) approach was used to describe the interactions between ions and valence electrons.67 The plane wave energy cutoff was set to 400 eV. The structures were optimized to achieve ion forces less than 10-3 eV·Å-1. A 6 × 6 × 1 Monkhorst-Pack k-point mesh 7


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was used.68 The van der Waals interactions were described using the Grimme DFT-D3 method.69 After

geometry

optimization

at

0

K,

the

156-atom

(BA)2PbI470

and

252-atom

(BA)2(MA)2Pb3I1032 systems were heated to 300 K by repeated velocity rescaling. Then, 3 ps adiabatic MD trajectories were generated with a 1fs time step, and the NA couplings between HOMO and LUMO were computed. The 3 ps NA Hamiltonian was iterated 4 times and 10 ps NAMD trajectories were obtained using the first 1000 geometries as initial conditions. The PYXAID software package was used, as described in detail in Refs. 44, 45 The time-domain method used in the present work uses standard DFT and includes explicitly inelastic and elastic electron-phonon scattering, the latter causing decoherence between the ground and excited electronic states. Excitonic effects requiring the Bethe-Salpeter theory on top of DFT are not included due to extremely high computational cost. So far, the timedependent version of the Bethe-Salpeter theory was applied only to systems with fixed nuclei.71, 72

The chosen approach incorporates electron correlation effects implicitly in the density

functional. Currently, it is the most rigorous ab initio method available for modelling the electron-phonon relaxation dynamics in the time-domain and including coupling to phonons. The 2D (BA)n(MA)n-1PbnI3n+1 perovskites can be viewed as derivatives of the 3D MAPbI3 perovskite that is cut along the (110) plane, and in which the MA cations are partially or totally substituted by longer BA cations within the cuts.32 A tilted [PbI6]4- octahedron is replicated periodically in the x-y plane, while n refers to the number of [PbI6]4- layers replicated in the z-

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direction. The slabs containing n layers of [PbI6]4- are separated by organic cations, such as BA, with long molecular chains. For n>1, the inorganic framework embeds smaller cations, such as MA, similarly to the 3D perovskites. No such cations are present for n=1. Figure 1 shows the optimized structures of (BA)2PbI4 and (BA)2(MA)2Pb3I10 and representative structures taken from the MD trajectories at room temperature. The averaged I-Pb bond length is 3.245 Å in the optimized (BA)2PbI4 monolayer, in agreement with the experimental value of 3.265 Å.32, 70 The bond shortens to 3.191 Å in the (BA)2(MA)2Pb3I10 trilayer due to intralayer interactions. Similarly to the I-Pb bonds, the average distance between the inorganic slabs separated by the BA chains is smaller in the tri-layer perovskite, 7.100 Å, compared with 7.428 Å in the monolayer perovskite. Overall, (BA)2(MA)2Pb3I10 is a more strongly bound structure than (BA)2PbI4 with shorter bond lengths. The canonically averaged I-Pb bond lengths are greater than those in the optimized structures: 3.284 Å in (BA)2PbI4 and 3.223 Å in (BA)2(MA)2Pb3I10. I-Pb bond fluctuations influence chemical bonding of the inorganic cage that supports the charge carriers. Rearrangements of MA cations in the tri-layer distorts the inorganic I-Pb cage. The inorganic slabs separated by BA cations can slide against each other, since the BA molecular chains are quite flexible. In order to validate the classical path approximation, we compared changes in the optimized geometries for the ground and excited electronic states with thermally induced atomic 9


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fluctuations in the ground state. The excited state geometry was obtained using the δ-SCF approach, in which an electron was promoted across the bandgap. The averaged I-Pb bond lengths for the excited state were 3.253 Å in (BA)2PbI4 and 3.205 Å in (BA)2(MA)2Pb3I10. The changes from the ground state geometries were only 0.009 Å and 0.014 Å, respectively. These are nearly an order of magnitude smaller than thermal geometry fluctuations, validating the approximation. Even the thermal expansion, characterized by the difference between the optimized ground state geometry and the canonically averaged ground state bond lengths, 0.039 Å in (BA)2PbI4 and 0.032 Å in (BA)2(MA)2Pb3I10, was much more significant than the difference between the ground and excited state geometries. All components of the hybrid perovskite materials exhibit notable anharmonicity and dynamic disorder, including both the inorganic framework and organic cations. Miyata et al. considered the effect of electron and hole polarons on the inorganic framework of CH3NH3PbBr3.73 The hole polaron structure had a more significant variation of the Br-Pb bond lengths and Pb-Br-Pb bond angle than the electron polaron structure. In particular, the average Br-Pb bond lengths shorten by 0.04 Å in the hole polaron and elongate by 0.01 Å in the electron polaron. The corresponding changes in the average Pb-Br-Pb angles are +5o and -2o, respectively. Miyata et al. attributed such changes to the dominant involvement of the coupled Pb-Br stretching and Pb-Br-Pb bending modes in the polaron stabilization. Rotation of the organic groups38, 74, 75 has a notable impact on the perovskite electronic and geometric structure, since they generate strong electric fields.76 Quarti et al.74 showed that MA cations rotate more slowly 10


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in the tetragonal phase of MAPbI3 than in the cubic phase due to strong hydrogen bond network in the former case hindering cation rotation. The reported rotation angles ranged up to 57o-89o in the tetragonal phase and 349o in the cubic phase.74 The picosecond timescale of reorientation of the MA cation in the cubic phase approaches that of a freely rotating MA ion, while for the tetragonal phase the rotation is increasingly arrested, with MA cations distributed over eight disordered states.75 Our previous study shows that rotation of the MA cations in cubic MAPbBr3 can change the bandgap by as much as 0.33 eV.38

Figure 1. Simulation cell showing geometry of (a) (BA)2PbI4 and (b) (BA)2(MA)2Pb3I10. The structures at 0 K and 300 K demonstrate the importance of thermal atomic fluctuations that are responsible for electron-phonon interactions. The organic MA cations, present inside the Pb-I inorganic framework of

(BA)2PbI4, create additional electron-phonon coupling.

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Figure 2 shows the projected density of states (PDOS) of (BA)2PbI4 and (BA)2(MA)2Pb3I10 split into the BA, MA, I and Pb contributions. Similarly to the 3D MAPbI3 perovskite,34 the HOMO of the 2D perovskites is composed of I and Pb atomic orbitals, while the LUMO arises primarily from Pb atoms. The organic groups do not contribute to the band edge states, and hence, they have no direct influence on the NA electron-phonon coupling and electron-hole recombination. The HOMO and LUMO of (BA)2PbI4 are separated by a wide bandgap of 2.25 eV. This value is in agreement with the previous DFT calculations21 and the experimental value of 2.70 eV.27 The bandgap reduces to 1.71 eV in the (BA)2(MA)2Pb3I10 trilayer due to weaker quantum confinement, also showing agreement with the previous DFT calculations32 and the experimental value of 2.17 eV.27 The calculated bandgaps are close to the experimental emission energies of 2.4 eV and 1.9 eV for the mono- and tri-layers, respectively.25 Typically, smaller bandgaps are associated with faster non-radiative electron-hole recombination, since they bring the electronic and vibrational quanta closer to resonance. However, it is not the case with the current 2D perovskites, among which the tri- and bi-layer systems show longer excited state lifetimes than the mono-layer.25 This counter-intuitive behavior requires an explanation.

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Figure 2. Projected density of states (PDOS) of (a) (BA)2PbI4 and (b) (BA)2(MA)2Pb3I10 calculated using the optimized geometries, and separated into contributions from BA, MA, I and Pb. Note that monolayer (BA)2PbI4 has no MA cations. Zero energy is set to the Fermi level. LUMO is formed by Pb atomic orbitals, and HOMO arises from I and Pb atomic orbitals, for both (BA)2PbI4 and (BA)2(MA)2Pb3I10. Due to quantum confinement, the (BA)2PbI4 monolayer has a larger bandgap than the (BA)2(MA)2Pb3I10 trilayer.

The nonradiative electron-hole recombination rate depends strongly on the magnitude of the NA coupling between the initial and final states. Since intraband relaxation in semiconductors occurs on sub-picosecond timescales,53, 55, 61 it is assumed that the electron and hole have relaxed to LUMO and HOMO prior to the recombination. The NA coupling depends on the overlap between HOMO and LUMO, whose charge densities are shown in Figure 3 for 13


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both (BA)2PbI4 and (BA)2(MA)2Pb3I10. The HOMO and LUMO are confined within a single inorganic layer in (BA)2PbI4, and therefore, exhibit strong overlap, Figure 3a. In comparison, the orbitals of (BA)2(MA)2Pb3I10 have the freedom to localize on different inorganic layers. Indeed, the HOMO shows stronger localization on the middle layer, while the LUMO prefers the outer layer, Figure 3b. This behavior leads one to expect a smaller NA coupling for (BA)2(MA)2Pb3I10 vs. (BA)2PbI4. However, it is not the case. The corresponding values are 0.41 meV and 1.49 meV, Table 1. Thus, both smaller bandgap and larger NA coupling indicate that the non-radiative electron-hole recombination should be faster in (BA)2(MA)2Pb3I10, but the experiments show the opposite result.25

Figure 3. HOMO and LUMO charge densities for (a) (BA)2PbI4 and (b) (BA)2(MA)2Pb3I10. The electron is localized on Pb atoms, and the hole is localized on both I and Pb atoms.

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Table 1. Experimental bandgaps,27 and calculated average absolute non-adiabatic coupling, puredephasing time, homogeneous linewidth, and nonradiative electron-hole recombination time for (BA)2PbI4 and (BA)2(MA)2Pb3I10.

(BA)2PbI4

2.70

non-adiabatic coupling (meV) 0.41

(BA)2(MA)2Pb3I10

2.17

1.49

bandgap (eV)

puredephasing (fs) 8.1

homogeneous linewidth (meV) 80

4.7

138

recombination (ps) 645 1035

Electron-vibrational interactions give rise to inelastic and elastic electron-phonon scattering. Inelastic scattering transfers the electronic energy to phonons, while elastic scattering destroys coherence between initial and final states. The inelastic scattering is characterized by the NA coupling matrix element −ħ ∇ ∙

, which depends on the phonon velocity and the

overlap of the initial and final electronic wavefunctions. Elastic scattering leads to puredephasing in the optical response theory.50 The inverse of the pure-dephasing time gives the homogeneous optical linewidth, 50-52 which can be obtained from luminescence measurements on individual chromophores. Averaging over a chromophore ensemble adds inhomogeneous linewidth. In order to characterize the phonon modes that couple to the electronic transition, we computed Fourier transforms (FT) of autocorrelation functions (ACF) of fluctuations of the 15


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electronic energy gaps along the MD trajectories, Figure 4. Low frequency vibrations participate in the nonradiative decay in (BA)2PbI4, compared to the higher and broader range of frequencies that couple to the relaxation in (BA)2(MA)2Pb3I10. First of all, the differences arise because the tri-layer has MA cations embedded into the Pb-I lattice, while the mono-layer does not. Although MAs do not contribute to the HOMO and LUMO, Figure 3, they distort the Pb-I lattice and create strong electric fields influencing the wave-functions. Second, the Pb-I lattice is stiffer and thicker in the tri-layer than in the mono-layer, also generating higher frequency modes. Participation of higher frequency phonons in the electron-phonon relaxation dynamics is the key result, rationalizing the larger NA coupling in (BA)2(MA)2Pb3I10 in spite of the smaller HOMOLUMO overlap, Figure 3, and leading to faster coherence loss, Table 1. The two peaks in the (BA)2PbI4 spectral density, Figure 4a, can be assigned to the Raman-active 120 cm-1 mode stemming from I-Pb stretching motions32 and 220 cm-1 torsional motions of the BA cations.77 These two phonons play a much weaker role in (BA)2(MA)2Pb3I10, which exhibits multiple higher frequencies, Figure 4b. The peaks in the 200-400 cm-1 region can be attributed to the torsional motions of the MA cations.78, 79 The dominant peak at 450 cm-1 can be assigned to the Pb3I10 vibration that is Raman active in (BA)2(MA)2Pb3I10.32 The peaks at 500 cm-1 and above are likely overtones of the lower frequency vibrations. The presence of multiple frequencies in the vibrational influence spectrum accelerate decoherence. To demonstrate the extent of fluctuation of the inorganic framework in the considered

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systems, we selected representative Pb-I bonds and Pb-I-Pb and I-Pb-I angles, and provided their optimized values as well as snapshots from the MD trajectory, Table S1 of Supporting Information. For instance, the Pb(1)-I(1)-Pb(2) angle in the mono-layer decreases by up to 22o at room temperature with respect to the optimized geometry. The corresponding angle in the trilayer decreases by as much as 46o. The Pb(1)-I(1) bond length of the monolayer fluctuates within the range from -0.067 Å to +0.369 Å around the optimized value. The corresponding fluctuation in the tri-layer ranges from -0.102 Å to +0.729 Å. The bond length and angle fluctuations are notably larger in the tri-layer than the monolayer, indicating that the electron-phonon interactions and anharmonicity are larger for the tri-layer.

Figure 4. Fourier transforms of autocorrelation functions for the fluctuations of the HOMO-LUMO gap in (a) (BA)2PbI4 and (b) (BA)2(MA)2Pb3I10. The higher and broader range of frequencies involved in tri-

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layer (BA)2(MA)2Pb3I10 lead to stronger electron-phonon coupling and faster phonon-induced loss of electronic coherence compared with monolayer (BA)2PbI4, Table 1 and Figure 5.

Compared to the monolayer (BA)2PbI4, the tri-layer (BA)2(MA)2Pb3I10 contains additional MA ions which carry positive charge and large dipole moment. The ions create strong electric fields that couple to electrons and holes, and are free to rotate inside the inorganic Pb-I cage. The fields are so significant that if the ions are not allowed to rotate, they can form para- and ferroelectric phases.80, 81 Vibrational motions of the MA cations contribute to the electron-phonon scattering through the electric fields they generate. In addition, the MA cations distort the inorganic Pb-I cage. This coupling of MA to Pb-I provides another electron-phonon scattering mechanism involving MAs. The decoherence time can be computed as the pure-dephasing time of the optical response theory50 according to the second-order cumulant approximation: () = − Here,

()

ħ

(

)!

(1)

is the unnormalized autocorrelation function (ACF) of the phonon-induced

fluctuation of the energy gap "# (), between electronic states i and j, defined as ()

= 〈"# ( )"# ( − )〉

(2)

The decoherence functions are shown in Figure 5, and the unnormalized ACF are presented in the insert. The pure-dephasing times, τ, Table 1, were obtained by fitting the data to a Gaussian, exp [−0.5(−//)0 ] . They are much shorter than the electron-hole recombination times,25 requiring incorporation of decoherence into NAMD.47, 58, 82 18


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Analysis shows that decoherence is determined by the initial amplitude and decay of the ACF.83, 84 A greater initial value and a more asymmetric ACF favor faster dephasing.83, 84 The square root of the initial value of the unnormalized ACF,

(0),

presents the magnitude of the

phonon-induced fluctuation of the electronic energy gap, characterizing inhomogeneous linebroadening. The gap fluctuations are 79.8 meV and 152.7 meV for (BA)2PbI4 and (BA)2(MA)2Pb3I10, respectively. The Planck’s constant divided by the pure-dephasing time gives the homogeneous contribution to the linewidth. The homogeneous linewidths shown in Table 1 are similar to the inhomogeneous contributions. The room temperature spectra reported in Figure 4a and Figure 5b of the experimental work25 demonstrate that the tri-layer perovskite has a broader emission than the mono-layer. Assuming that the single flakes used to obtain the experimental spectra represent individual emitters, the experimental linewidths represent homogeneous broadening. The calculated homogeneous linewidths agree very well with the experimental data.

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Figure

5.

Pure-dephasing

functions

for

the

HOMO-LUMO

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transition

in

(BA)2PbI4

and

(BA)2(MA)2Pb3I10. The decay time represents elastic electron-phonon scattering responsible for coherence loss. The pure-dephasing times, Table 1, are obtained by Gaussians fitting. The inset shows the unnormalized autocorrelation functions, whose initial values give the bandgap fluctuation squared. Typically, the greater the initial value, the faster the dephasing.84

Figure 6 presents evolution of the excited state population in the two systems under investigation. Because PBE somewhat underestimates the bandgaps, they were scaled to the experimental values,25, 27 reproduced in Table 1. The nonradiative electron-hole recombination times, obtained using the short-time linear expansion of exponential decay, 2() = exp(−/ /)≈ 1 − //, and summarized in Table 1, show excellent agreement with experiment.25 Slow 20


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recombination reduces energy losses and improves solar cell performance. The slower electronhole recombination calculated for (BA)2(MA)2Pb3I10 can explain its higher measured26 photoelectric efficiency compared with (BA)2PbI4. The electron-hole recombination time depends on the energy gap, NA coupling and quantum coherence. (BA)2PbI4 exhibits larger gap and smaller coupling than (BA)2(MA)2Pb3I10, Table 1, suggesting that the lifetime should be longer. The opposite result seen in the experiments.25 Our calculations explain the experimental finding by faster decoherence, which slows down quantum dynamics, as exemplified by the quantum-Zeno effect.85 Decoherence is a time-domain analog of the Franck-Condon and Huang-Rhys factors, defined in the energy domain.82,

86

Faster decoherence corresponds to a smaller Franck-Condon factor and lower

transition rate.82

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Figure 6. Electron-hole recombination dynamics in (BA)2PbI4 and (BA)2(MA)2Pb3I10. Tri-layer (BA)2(MA)2Pb3I10 shows slower decay compared to monolayer (BA)2PbI4 due to the shorter coherence time, despite larger NA coupling and smaller bandgap, Table 1.

In order to establish that the PL lifetimes are limited by nonradiative rather than radiative decay, we calculated the radiative lifetimes. The Einstein coefficient for spontaneous emission, 40 , between states 1 and 2 relates the oscillator strength, 50 , and the lifetime, /0 , of the specified transition: 40 =

: A 67 89 9

5 ;< => ? @ A 0

(3)

Here, B is the transition frequency, and , C , D: , and E are the fundamental constants. The state

degeneracies are F = F0 = 1 in the current system, because it has no symmetry due to thermal atomic fluctuations. The emission lifetime is inverse of the Einstein coefficient. The calculated 22


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radiative recombination times for the mono- and tri-layer perovskites are 2.47 ns and 5.08 ns, respectively. They are longer than the nonradiative electron-hole recombination times, Table 1 and Figure 6, indicating that nonradiative electron-hole recombination constitutes the main process responsible for luminescence decay, supporting our interpretation of the experimental data. The current work focuses on electron-hole recombination at ambient temperature, which is most relevant in applications. Electron-phonon scattering slows down with decreasing temperature, and as a result, luminescence lifetime increases. The experimental data reported in Figure 2 of ref.25 confirm this trend. In conclusion, we have reported the first ab initio time-domain study of nonradiative electron-hole recombination in 2D layered Ruddlesden-Popper halide perovskites, which hold great promise for photovoltaic and optoelectronic applications. The recombination constitutes the major route for charge and energy losses, limiting photon-to-electron conversion efficiency in the 2D perovskite solar cells. The simulations rationalize why the tri-layer perovskite shows a longer excited-state lifetime compared to the mono-layer material, even though the tri-layer has a smaller gap and stronger electron-phonon coupling. It is elastic rather than inelastic electronphonon scattering that provides the key to explaining the experiments. Stronger elastic scattering leads to faster loss of coherence within the electronic subsystem, slowing down quantum dynamics. Decoherence is sub-10fs in both materials, resulting in excited state lifetimes that extend into nanoseconds. The calculated coherence times are directly related to the homogeneous 23


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linewidths, showing excellent agreement with the experiment. We predict that the heterogeneous contributions to the linewidths are similar to the heterogeneous contributions. The energy gap is smaller in the tri-layer than the mono-layer due to decreased quantum confinement effect. The electron-phonon coupling is stronger in the tri-layer because, in contrast to the monolayer, it contains organic MA groups embedded into the inorganic Pb-I lattice. The fast motion of the MA species enhances the coupling. The thicker Pb-I layer also supports high frequency modes that increase the overall electron-phonon coupling in the tri-layer. The reported findings highlight the importance of quantum coherence in the photoexcitation dynamics of 2D perovskites and provides important mechanistic insights into charge carrier recombination that determines performance of 2D perovskite optoelectronic devices and solar cells.

Notes The authors declare no competing financial interest.

Acknowledgements Z. Z., W.-H. F. and R. L. acknowledge funding from the National Science Foundation of China, grant Nos. 21573022, 21688102, 21590801, and 21421003. R. L. acknowledges financial support by the Recruitment Program of Global Youth Experts of China, the Beijing Normal University Startup, and the Fundamental Research Funds for the Central Universities. O. V. P. is grateful to the 1000 talents plan for supporting his visit to Beijing Normal University. M. V. T. 24


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and O. V. P. acknowledge support of the U.S. National Science Foundation, award Number CHE1565704.

Supporting Information Available: Representative Pb-I bond lengths and Pb-I-Pb, I-Pb-I bond angles taken from the optimized geometries and MD trajectory snapshots. The material is available free of charge via the Internet at http://pubs.acs.org.

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Rapid Decoherence Suppresses Charge Recombination in Multi

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