Mixed Cs and FA Cations Slow ElectronâHole Recombination in

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Energy Conversion and Storage; Plasmonics and Optoelectronics

Mixed Cs and FA Cations Slow Electron-Hole Recombination in FAPbI3 Perovskite by Time-Domain Ab Initio Study: Lattice Contraction versus Octahedral Tilting Lu Qiao, Xueqin Sun, and Run Long J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b03729 • Publication Date (Web): 25 Jan 2019 Downloaded from http://pubs.acs.org on January 25, 2019

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The Journal of Physical Chemistry Letters

Mixed Cs and FA Cations Slow Electron-Hole Recombination in FAPbI3 Perovskite by Time-Domain Ab Initio Study: Lattice Contraction versus Octahedral Tilting Lu Qiao,1 Xueqin Sun,2 Run Long1* 1College

of Chemistry, Key Laboratory of Theoretical & Computational Photochemistry of

Ministry of Education, Beijing Normal University, Beijing, 100875, P. R. China 2School

of Environmental and Material Engineering, Yantai University, Yantai 264005, P. R. China

ABSTRACT: Using time domain density functional theory combined with nonadiabatic (NA) molecular dynamics, we show that electron-hole recombination takes sub-nanosecond in FAPbI3, showing excellent agreement with experiment. Cs doping retards charge recombination by factors of 1.1 and 3.1, due to lattice contraction and octahedral tilting, respectively. Lattice contraction decreases the NA coupling and increases the coherence time arising from the suppressed atomic fluctuations, slightly slowing recombination because the two factors have an opposite influence on quantum transition. In contrast, octahedral tilting simultaneously decreases the NA coupling, thanks to the reduced overlap between Pb and I orbitals, and coherence time, extending the excited-state lifetime over one nanosecond. Our simulations provide a mechanistic understanding for delayed charge losses in the mixed Cs and FA system, suggesting a rational strategy to improve perovskite solar cell performance.

*

Corresponding author, E-mail: [email protected] 1

ACS Paragon Plus Environment

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TOC only

2


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In the past few years, hybrid organic-inorganic perovskites (HOIPs) have become the most promising photovoltaic materials due to their excellent electronic and optical properties, including high optical absorption coefficient,1 long carrier diffusion length,2 low trap densities3 and small exciton binding energy.4 Since the first use of perovskite as light absorber in dye-sensitized solar cells in 2009,5 the power conversion efficiency (PCE) increased from 3.8%5 to 23.3%6 within several years. The advanced properties extend their applications to light-emitting diodes,7 lasers,8 and sensors.9 Most of previous research focuses on the methylammonium lead iodide (MAPbI3) because its band gap of 1.52 eV10 falls into the optimum range of 1.4-1.6 eV11 for a single junction photovoltaic solar cell. MAPbI3 perovskite suffers from, however, the inferior tolerance to heat and moisture, leading to the degradation during the fabrication and devices operation.12 Compared with the MAPbI3, formamidinium lead iodide (FAPbI3) exhibits an ideal band gap for light absorption of 1.45 eV,13 and provides superior stability and longer carrier lifetime than MAPbI3.14 Although FAPbI3 is considered to be a promising candidate for light harvester, the widely used pure α-FAPbI3 can readily transform into a non-perovskite hexagonal

-phase with a large band gap.10 This transformation occurs

very rapidly at room temperature.15 Additionally, all inorganic Cs-based perovskites are difficult to fabricate homogeneous polycrystalline films, leading they to exhibit low current efficiency.16 Meanwhile, the large band gap, with the CsPbBr3 of 2.36 eV17 and CsPbI3 of 1.77 eV,18 further limits the application of Cs-based perovskites for photovoltaic solar cells.

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In order to simultaneously increase the stability of

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materials and harvest a wider range of

solar spectrum, mixing two or more monovalent cations (MA+, FA+, Rb+, Cs+) shows great potential to merge the advantages of the all-inorganic and HOIPs into one material.19-21 Park at al. showed that FAPbI3 doping with Rb+ can significantly prevent phase transition from ideal α -phase to

-phase and achieve a high power conversion efficiency of 17.16%, retaining the high

performance for 40 days under the ambient environment.22 Experiments demonstrated that replacing partial FA cations with Cs ions can achieve desirable thermodynamic stability23 and resistance to humidity24 with favorable optical and electrical properties,23 due to carrying the advantages of Cs based all-inorganic perovskites, including high stability, narrow emission spectral spectrum and photoluminescence quantum yield.23,24 Zhao at al. illustrated that incorporation of Cs+ into FAPbI3 can reduce trap density, improve stability and power conversion efficiency,25 with only a little decomposition in susceptibility to light exposing to air over one month.26 Recently, Prasanna et al. demonstrated that

Cs doping preferred to induce

octahedral tilting (OT) in FAPbI3 while lattice concentration (LC) in FASnI3,27 leading to an enhanced solar cell performance and extended excited-state lifetime. In order to save the computational cost, it is reasonable to separately investigate two effects on the nonradiative electron-hole recombination in one material. Because FAPbI3 is much more popular and shows higher power conversion efficiency than that of FASnI3,28,29 and thus we choose FAPbI3 to study the influence of LC and OT induced by Cs doping on the excited-state dynamics and aim to rationalize the enhanced photon-to-electron conversion efficiency reported experimentally.25 4


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Inspired by the experimental works,27,30 we report an ab initio atomistic time-domain study of LC and OT induced by Cs doping on electron-hole recombination in FAPbI3 and provide mechanistic understanding for reduced charge and energy losses. The study demonstrates that nonradiative electron-hole recombination in pristine FAPbI3 occurs within sub-nanosecond, agreeing well with experiment.30 Both LC and OT change the band gap slightly compared to the pristine system, leading to little influence on the electron-hole recombination. LC reduces nonadiabatic (NA) electron-phonon coupling and prolongs coherence time stemming from the suppressed nuclear motions, they have opposite influence on the electron-hole recombination, extending the lifetime only by a factor of 1.1. OT simultaneously reduces NA coupling and shortens coherence time due to the decreased overlap of electron and hole wave functions and fast atomic fluctuations, further extending the lifetime to several nanoseconds. The simulated results rationalize the effect of structure distortion induced by mixing two sizes of ions on electron-hole recombination at atomistic level and provide guidelines to improve the performance of the perovskite solar cells. The

NA

molecular

decoherence-induced

dynamics

surface

(NAMD)31

hopping32

technique

simulations

are

implemented

performed within

the

by

the

real-time

time-dependent Kohn-Sham density functional theory (DFT).33-36 The lighter and faster electrons are described quantum mechanically, and the heavier and slower nuclei are treated semi-classically. Decoherence is needed here because the decoherence time is extremetly shorter than the nonradiative electron-hole recombination time.30 The approach has been applied to 5



investigate photoexcitation

dynamics in a broad range of systems,37-39, including MAPbI3

containing grain boundary,37 forming localized charge,40 inorganic perovskite quantum dots,41 and two-dimensional Ruddlesden-Popper perovskite.42 Very recently, the spin-orbit couplings were implemented into NAMD algorithm and showed a significant influence on intraband charge relaxation in MAPbI3.43 Geometry optimization and adiabatic MD are performed using the Vienna ab initio simulation package (VASP),44 with the projected-augmented wave method for describing the electron-ion interactions45 and the Perdew-Burke-Ernzerhof

functional46 under the generalized

gradient approximation for treating the electron exchange-correlation energy.46 Grimme DFT-D3 approach is used to describe the van der Waals interactions within the systems under investigation.47 The plane-wave basis energy cutoff is set to 400 eV with a 4× 4 ×4 Monkhorst-Pack k-mesh

48

for geometry optimization and electronic structure calculations. The

geometry optimization stopped until the ion forces per atom are less than 10-3 eV · Å-1. After relaxing the geometry at 0 K, all the systems were heated up to 300 K using velocity rescaling. Then, a 4 ps adiabatic MD simulation is employed at Γ point in the microcanonical ensemble with a 1 fs atomic time-step. Furthermore, all geometries of the 4 ps adiabatic MD trajectories were chosen as initial conditions for NAMD simulation of electron-hole recombination using the PYXAID package.35,36 A detailed description of the method and its implementation can be found elsewhere. 35,36 The cubic phase of α-FAPbI3 with the optimized lattice constant of 6.40 Å, in agreement with experimental value of 6.36 Å,49 has been used to create a (2 × 2 × 2) supercell (Figure 1a). 6


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Replacing 25% FA with Cs cations in the pristine FAPbI3, the LC system is constructed via contracting lattice constant by 0.1 Å (Figure 1b), and the OT system is obtained by tilting the [PbI62-] octahedra with titling angle of 12° in the ab plane (Figure 1c). Compared the optimized geometries with the structures at 300 K, we observed that the geometry of LC system occurs a minor deformation while OT experiences a significant distortion upon heating related to the pristine system. For instance, the average length of Pb-I bond increases from 3.253 Å at 0 K to 3.302 Å at 300 K in the pristine FAPbI3, with an elongation of 0.049 Å. In the case of LC system, the average Pb-I bond length is 3.125 Å at 0 K while it becomes 3.149 Å at 300 K, taking place a smaller increase of 0.024 Å compared with the pristine system. While the OT system shows a larger increase of 0.072 Å in the average Pb-I bond length, from the 3.201 Å at 0K to 3.273 Å at 300K. The larger bond length variation suggests stronger atomic displacements, leading to rapid decoherence, Table 1.

7



Figure 1. Optimized geometry of (a) FAPbI3, Cs doping induced (b) lattice contraction (LC) and (c) octahedral tilting (OT). The LC is constructed by contracting the lattice constant of the pristine FAPbI3 from 6.40 Å to 6.30 Å, while OT is created by titling the Pb-I-Pb angle with 12° in the ab plane.

In addition to the average bong length, the canonically averaged standard deviation of and are calculated, because they constitute the

Pb and I atoms by the equation

initial and final state for electron-hole recombination.50 Here,

represents the position of atom

i at time t along the 4 ps adiabatic MD trajectories. Then, we average the standard deviations of Pb and I atoms in all systems. Figure 2 shows that the standard deviations in the positions of Pb and I atoms decrease from the OT system, to the pristine FAPbI3, to the LC system. Larger atomic fluctuations are responsible for shorter quantum coherence time, Table 1. This 8


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observation is in agreement with the above analysis of change in the averaged Pb-I bond length.

Figure 2. Standard deviations in the positions of Pb and I atoms in the pristine FAPbI3, LC, and OT.

Figure 3 shows the projected density of states (PDOS) of the pristine FAPbI3, LC and OT systems. The PDOS is splitting into contributions of FA, Cs, Pb, and I, respectively. Apparently, both FA and Cs cations do not contribute to the band edge states, agreeing with the previous calculations.50 They have no direct influence on the electron-hole recombination but affect it through an indirect way, such as perturbing the motions of inorganic framework. The band edge states are composed of Pb and I atoms. Shown in part a-c of Figure 3, HOMO and LUMO are composed primarily of I and Pb atoms, respectively, which are separated by a wide band gap of 1.41 eV. the excess electronic energy is accommodated by phonons during the nonradiative charge recombination. Because of the smaller overlap between p-p orbitals than that of s-p orbitals,27 geometry distortion will affect the charge distribution of HOMO more than that of 9



LUMO. Thus, LC will increase the overlap between I 5p and Pb 6s orbitals, lifting the HOMO energy. In contrast, OT decreases Pb-I-Pb angles and reduces the overlap between I 5p and Pb 6s orbitals, pushing down the VBM. As a result, LC decreases the band gap to 1.38 eV, while OT increases the band gap to 1.53 eV. The change trend in band gap is consistent with the previous experimental observations.39 The band gaps of the three systems are close to each other. It is expected that the influence of band gap on charge recombination is insignificant because electron-hole recombination rate is linearly proportional to energy gap according to gap laws.51

Figure 3. The projected density of states (PDOS) of the optimized structure of (a) FAPbI3, (b) LC, and (c) OT. Zero energy is set to the Fermi level. Lattice contraction decreases the band gap, while octahedral tilting increases it.

In addition to the band gap, NA electron-phonon coupling affects nonradiative electron-hole recombination significantly, which depends on the mixing of wave functions between the HOMO and LUMO, , and the nuclear velocity dR/dt. Generally, larger HOMO-LUMO 10


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overlap and stronger atomic fluctuations create stronger NA electron-phonon coupling. Figure 4 shows that in all of systems the HOMO and LUMO are primarily supported by I and Pb atoms, respectively, consistent with the PDOS analysis. Such charge distribution reduces the HOMO-LUMO overlap, decreases NA coupling and facilitates long-lived excited-state lifetime.30 Compared the pristine FAPbI3 (Figure 4a) with LC (Figure 4b) system, the charge densities of HOMO and LUMO show similar distribution. The difference in NA electron-phonon coupling mainly arises from the nuclear fluctuations. In addition to the reported Pb-I bond length variation, lattice free volume can reflect the trend of the I and Pb moving in the inorganic cages of the two systems, which is obtained by excluding the ions’ total volumes from the supercell volume.52 The calculated volume of the pristine FAPbI3 and LC supercell is 2097.15 and 2000.38 Å3, and the whole ion volumes of FAPbI3 and LC is calculated to be 1465.18 and 1435.24 Å3 based on the ionic radii of FA+, Cs+, Pb2+, and I− are 1.88, 2.17, 2.53, 1.16, and 2.2 Å,53,54 respectively. As a result, the free volume of FAPbI3 and LC are 631.97 and 565.14 Å3, respectively. The smaller free volume of LC indicates the suppressed atomic motions, decreasing the NA electron-phonon coupling. In addition, considering the weak polarity of Cs cations compared with FA cations, the Cs-doped system will obtain smaller NA coupling because of reduced inorganic framework distortion. Figure 4c demonstrates that in the OT system the charge density of HOMO is decreased with leaving LUMO largely unchanged, and thus HOMO-LUMO overlap is reduced and the NA coupling is decreased. Shown in Table 1, the NA electron-phonon coupling exhibits a decreased trend as the following order: FAPbI3 > LC > OT, suggesting that 11



wave function mixing primarily contributes to the NA coupling against the atomic fluctuations.

Figure 4. HOMO and LUMO charge densities of (a) FAPbI3, (b) LC and (c) OT, respectively.

Participation of the phonon modes is key to the recombination: they couple to the electronic degrees of freedom and promote the nonradiative electron-hole recombination. Figure 5 presents the spectral densities obtained from Fourier transforms (FTs) of the fluctuations for the HOMO-LUMO band gap. Only low-frequency vibrations participate in the nonradiative decay in all three systems. Particularly, these modes below 100 cm-1 drive primarily the recombination, and the intermediate frequencies ranging from 100 cm-1 to 200 cm-1 contribute secondarily. Figure 5a shows that in the pristine FAPbI3 the major peak at 100 cm-1 can be ascribed to the 12


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bending and stretching motions of the Pb-I bond.55 This mode creates the largest NA electron-phonon coupling. The peak around 33 cm-1 is related to the octahedra distortion.56 The minor peak at 166 cm-1 can be attributed to the libration of the organic cations.55 This mode associated with low intensity indicates that the organic cations have insignificant contribution to the NA coupling through an indirect way by perturbing the inorganic Pb-I backbones. Replacing patrt of larger FA ions with smaller Cs ions reduces the lattice constant and suppresses the atomic fluctuations (Figure 2), reflecting by the decreased intensity of the major peak at 100 cm-1, Figure 5b. At the same time, smaller Cs ions doping leads to significant octahedral tilting, giving rise to the enhancement of frequencies at 100 cm-1 that is responsible for creating the majority of NA coupling. Higher frequency and stronger intensity are responsible for rapid loss of quantum coherence. As a result, the quantum coherence time decreass in the sequence: LC > FAPbI3 > OT, Table 1.

13



Figure 5. Fourier transforms of autocorrelation functions for the HOMO−LUMO gap in (a) FAPbI3, (b) LC and (c) OT respectively.

The decoherence, known as pure-dephasing, whose times were computed in optical response theory using the second-order cumulant approximation:57 (1)

Here,

is the unnormalized autocorrelation function (un-ACF) of the thermal fluctuation

of the HOMO-LUMO energy gap,

.

is

computed by following formula (2)

The pure-dephasing times, τ , summarized in Table 1, are obtained by fitting the data in 14


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Figure 6to a Gaussian,

. The sub-10 fs pure-dephaisng times in the three

systems are much shorter than the electron-hole recombination times reported in the experiment,30,58 necessitating the incorporation of decoherence into the NAMD simulation.The un-ACF of the HOMO-LUMO energy gap fluctuations (inset of Figure 6), can explain the change trend in the pure-dephasing times in the three systems. According to the second-order approximation for the optical response theory,57

the dephasing function is computed by

integrating un-ACF. Generally, a larger initial value of un-ACF and its slower decay are responsible for a shorter dephasing time. Shown in the inset of Figure 6, the two un-ACFs of the pristine FAPbI3 and LC system decay on similar time scales, the differences in the pure-dephasing times originate from the un-ACF intial values,

. On the contrary, the

un-ACF of the OT system exbibits the largest and decays slower than the other two systems, leading to the shortest depahsing time. Rapid coherence favors slow quantum transiton as a manifestation of the Quantum-Zeno effect.59 In the limit of infinitely fast coherence, the quantum transiton stops.

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Figure 6. Pure-dephasing functions for the HOMO−LUMO transition in each system. The inset shows the unnormalized autocorrelation functions (ACF). The bigger the initial value and slower decay of the un-ACF, the faster the pure-dephasing.

Figure 7 shows the electron-hole recombination dynamics for the three systems. The nonradiative decay times, summarized in Table 1, are obtained using the short-time linear approximation to the exponential decay,

. We have scaled the

calculated band gap of 1.41eV of the pristine FAPbI3 to the experimental value of 1.45eV13 by adding a constant for simulating electron-hole recombination dynamics. The band gap of

other

two systems have been tuned by adding the same constant assuming that PBE functional exhibits 16


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identical error. Shown in Table 1, the small variation of band gaps in the three systems indicate that their influence on the recombination is insignificant. The calculated charge recombination time 520 ps for the pristine FAPbI3 falls in the middle of the 400-1400 ps experimental range.30,58 The slow recombination is beneficial for reducing charge and energy losses and are responsible for a high power conversion efficiency of perovskite solar cells.60 Upon lattice contraction and octahedral tilting induced by Cs doping, the recombination times reduce by factors of 1.1 and 3.1, respectively.

Compared to the pristine FAPbI3, the smaller NA

electron-phonon coupling in the C system competes successfully with the longer coherence time, extending the excited-state lifetime to 560 ps, Table 1. In contrast, the coherence time shortens to 6.7 fs with octahedral tilting. At the same time, the reduced HOMO-LUMO overlap and suppressed atomic motions decrease the NA coupling by a factor of 1.7, extending the excited-state lifetime to 1600 ps in the OT system.

17



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Figure 7. Nonradiative electron-hole recombination dynamics in the FAPbI3, LC and OT.

Table 1. Band gap, Average NA coupling, Pure-Dephasing Time, Nonradiative electron-hole recombination Time for FAPbI3, LC and OT.

Bandgap (eV)

NA coupling (meV)

Dephasing (fs)

Recombination (ps)

FAPbI3

1.45

1.16

7.2

520

LC

1.42

1.05

7.7

560

OT

1.57

0.69

6.4

1600

18


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In addition to nonradiative decay, radiative decay plays another key factor affecting perovskite optoelectronic properties, such as photoluminescence lifetimes. Therefore, we calculated the radiative lifetimes of the three systems according to the Einstein coefficient of spontaneous emission:61

. The emission lifetime is equal to the inverse of and

the Einstein coefficient. Here,

are fundamental constants. The state

degeneracies are 𝑔1 = 𝑔2 = 1 the three systems because they have no symmetry due to thermal fluctuations. 𝑓12 is the oscillator strength and v12 is the transition frequency between the states 1 and state 2, respectively. The calculated radiative recombination time for the FAPbI3 is 1.7 ns, consistent with experimental value.62

Upon geometry deformation, the radiative recombination

times increases to 2.2 and 6.3 ns for the LC and OT systems, respectively. Importantly, the radiative decays are significantly slower than the nonradiative electron-hole recombination (Table 1), suggesting that nonradiative electron-hole recombination is the major limitation in luminescence decay. In summary, we have simulated nonradiative electron-hole recombination in pristine FAPbI3 with and without Cs doping using ab initio nonadiabatic molecular dynamics. The simulations show that the electron-hole recombination in these materials takes place ranging from sub-nanosecond to over nanoseconds. We consider that smaller Cs ions substitution to the larger FA simultaneously lead to lattice contraction and octahedral tilting in the FAPbI3. Lattice contraction weakens the NA coupling while increases the coherence time slightly that have an opposite effect on the quantum transitions, slowing the recombination by a factor of 1.1. Octahedral tilting simultaneously decreases the NA coupling and coherence time, retarding the recombination by a factor of 3.1. The recombination is slow overall because the NA coupling is 19



small, sub-1.5 meV, and the coherence time is short, sub-10 fs. The coupling is weak and the coherence is short because electrons and holes are composed by Pb and I, respectively. The recombination is assisted primarily by low-frequency, sub-200 cm-1 modes, which further rationalize the small NA coupling magnitude. The NA coupling decreases as the suppressed atomic motions with lattice contraction because the coupling is proportional to the nuclear velocity, and which decreases further with octahedral tilting due to the reduced overlap between electron and hole wave functions. The obtained nonradiative electron-hole recombination time scales agree well with the experimental data. The simulations provide a mechanistic understanding for the influence of structural distortion with incorporation of dopants on nonradiative electron-hole recombination dynamics and suggest a new avenue to reduce energy loss in perovskite materials through doping to improve the photovoltaic efficiency.

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was supported by the National Science Foundation of China, grant Nos. 21573022 and 51861135101. R. L. acknowledges financial support by the Fundamental Research Funds for the Central Universities and the Beijing Normal University Startup.

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