Doping-Induced Rapid Decoherence Suppresses Charge

Letter Cite This: J. Phys. Chem. Lett. 2019, 10, 3433−3439

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Doping-Induced Rapid Decoherence Suppresses Charge Recombination in Mono/Divalent Cation Mixed Perovskites from Nonadiabatic Molecular Dynamics Simulation Zhaosheng Zhang and Run Long*

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College of Chemistry, Key Laboratory of Theoretical & Computational Photochemistry of Ministry of Education, Beijing Normal University, Beijing 100875, People’s Republic of China ABSTRACT: Experiment shows that solar cells based on FA0.75Cs0.25Pb0.5Sn0.5I3 carry a lower charge recombination rate and higher power conversion efficiency than those of FAPbI3 despite the fact that the former has a smaller band gap. However, the underlying mechanism remains unclear. Using nonadiabatic (NA) molecular dynamics, we demonstrate that low-frequency vibrations drive electron−hole recombination in pristine FAPbI3 occurring in about 1 ns, showing excellent agreement with experiment. Cs/Sn substitution to FA/Pb not only narrows its band gap by 0.3 eV but also delocalizes the electron wave function significantly, leading to enhancement of NA coupling. Importantly, doping accelerates quantum decoherence caused by increased atomic fluctuations. As a result, rapid decoherence prevails a small band gap and strong NA coupling, slowing charge recombination and extending the charge carriers’ lifetime to several nanoseconds. Our study reveals the importance of quantum coherence on quantum dynamics in perovskite materials and suggests a rational strategy to design high-performance perovskite solar cells.

A

transition in the structure, leading to extended charge carrier lifetimes.35 However, α-FAPbI3 is a thermodynamically unstable phase at room temperature that spontaneously converts to yellow δ-FAPbI3, losing its excellent optical and electronic properties.15,36 Experiments have shown that mixed cations and/or anions can suppress the α-to-δ-phase transition,37,38 enhance material stability,38 and improve solar cell performance. 37 For example, Wang et al.39 have demonstrated that the PEC of FA0.83Cs0.17Pb(I0.6Br0.4)3 solar cells achieves 18.3%, which sustains 80% efficiency over 3400 h under full sun illumination in ambient conditions. However, the large band gap (1.74 eV) of this mixed perovskite impedes further PEC improvement.40 Further replacing Pb with Sn not only lowers the toxicity but also reduces the band gap of FA0.75Cs0.25Pb0.5Sn0.5I3 to 1.22 eV, showing an enhancement of 19.1% PEC and an external quantum efficiency of >80% in the near-infrared regime.41 The notable improvement can be attributed to the extended excite-state lifetimes42 and suppressed nonradiative electron−hole recombination that constitute the main pathway for charge and energy losses.43 However, the underlying mechanism for that reduced charge recombination remains largely unclear and requires a comprehensive theoretical study in real time and at the atomistic level. Motivated by recent experiments,42,44−47 we investigate nonradiative electron−hole recombination in FAPbI3 (Figure 1a) and FA0.75Cs0.25Pb0.5Sn0.5I3 (Figure 1b) using a combina-

s promising optoelectronic and photovoltaic materials, hybrid organic−inorganic perovskites (HOIPs) have been widely used in a broad range of areas,1−7 including lightemitting devices,1,2 lasers,3 and solar cells.5−7 In particular, research on perovskite solar cells has progressed extremely rapidly compared to that on traditional silicon solar cells, promoting the power conversion efficiency (PCE) from an initial 3.814 to 24.2%8 in the past decade. Such great improvement originates from their excellent photoelectric properties, such as long carrier diffusion lengths,9−12 tunable band gaps with mixed cations and/or anions,13−15 and long excited-state lifetimes.10,16−20 Especially, long-lived charge carriers mean a low nonradiative charge recombination rate, whose mechanisms remain an active debate, including that collective rotations of dipolar MA cations enhance charge transport,21−23 protect hot carriers in halide perovskites,24,25 and convert detrimental deep defects into harmless shallow electronic states.26 The central idea focuses on rotationinduced geometry distortion. Furthermore, low-temperature solution processability27,28 and low-cost synthesis14,29 of HOIPs provide key advantages for large-scale commercialization. However, the classic perovskite MAPbI3 (MA = CH3NH3) suffers from significant stability issues upon exposure to light and a thermal environment30 and also the band gap of 1.65 eV31 is larger than the ideal band gap (1.34 eV) of a single cell with the maximum PCE, leaving a margin to approaching the Shockley−Queisser limit of 34%.32 Compared with MAPbI3, α-FAPbI3 (FA = HC(NH2)2) possesses a smaller band gap (1.50 eV) and has received persistent interest for the fabrication of solar cells.33,34 The reorientation of FA cations can also cause an order−disorder © 2019 American Chemical Society

Received: May 10, 2019 Accepted: June 3, 2019 Published: June 3, 2019 3433

DOI: 10.1021/acs.jpclett.9b01330 J. Phys. Chem. Lett. 2019, 10, 3433−3439

Letter

The Journal of Physical Chemistry Letters

is significantly shorter than electron−hole recombination, which occurs in several nanoseconds.42 Geometry optimization, adiabatic MD, and NA coupling were carried out by using the Vienna ab initio simulation package (VASP).56 The Perdew−Burke−Ernzerh (PBE) functional was used to describe electronic exchange− correlation interactions,57 and the projector-augmented wave (PAW) approach was used to treat ion−electron interactions.58 The plane wave energy cutoff was set to 400 eV with a 6 × 6 × 6 Monkhorst−Pack k-point mesh.59 The optimization stopped until ion forces were less than 10−3 eV·Å−1. The van der Waals interactions were described using the Grimme DFTD3 method with Becke−Johnson damping.60,61 After geometry optimization at 0 K, all three systems were heated to 300 K by repeated velocity rescaling for 2 ps. Then, a 6 ps trajectory was obtained in the microcanonical ensemble with a 1 fs atomic time step. The NA couplings were computed using the first 2000 geometries of the adiabatic MD trajectories, which were iterated six times to run 9 ps NAMD simulations by choosing all 2 ps trajectories as initial conditions. Figure 1 shows the optimized structures of FAPbI3 (Figure 1a) and Cs/Sn-doped FAPbI3 (Figure 1b) and representative MD snapshots at room temperature. For the doped case, 25% FA and 50% Pb were replaced by Cs and Sn, respectively, to match doping concentrations reported experimentally.42 Because the ionic radii follow the order Cs+ < FA+ and Sn2+ < Pb2+, replacing FA+ and Pb2+ with Cs+ and Sn2+ in the FAPbI3 leads to a notable geometry distortion in the inorganic Pb−I lattice, particularly on the octahedra conducting Cs dopants at 0 K (top panel of Figure 1b). For instance, the average I−Pb bond length is 3.182 in the optimized FAPbI3, agreeing well with the experimental value of 3.181 Å.33 The average I−Pb bond length increases to 3.206 Å after doping. The slight increase in bond length arises due to more free space allowing P and I atoms move away from each other. When the temperature is evaluated to 300 K, the distortion becomes much more significant at room temperature because of thermal fluctuations as well as the rotations of flexible FA cations in the Pb−I cage, in particular, for the Cs/Sn-doped case.35 The canonically averaged Pb−I bond length of the doped case is greater than that in the optimized structure, 3.395 vs 3.325 Å, reflecting the fact that nuclei dynamics have a stronger influence on the Cs/Sn-doped FAPbI3. Because Pb and I atoms constitute the band edges of lead halide perovskites and contribute to creating the majority of NA electron−phonon coupling,62,63 significant atomic fluctuations in the inorganic lattice increase coupling and accelerate the phonon-induced loss of quantum coherence. They have an opposite impact on the quantum transition dynamics. To validate the classical path approximation, we also calculated the average I−Pb bond lengths for the optimized geometries at the excited state using the δ-SCF approach. The values were 3.245 Å in the FAPbI3 and 3.304 Å in the doped FAPbI3.The changes between the excited-state and groundstate geometries at 0 K were 0.063 and 0.098 Å for the FAPbI3 and Cs/Sn-doped cases, respectively. In contrast, thermal fluctuations increase the difference to 0.143 Å for the FAPbI3 and 0.189 Å for the two cases, respectively. Apparently, thermal fluctuations give rise to much more significant geometry changes than light excitation, suggesting that the classical path approximation is reasonable for the present halide perovskite systems.

Figure 1. Simulation cell showing geometries of (a) FAPbI3 and (b) Cs/Sn-doped FAPbI3 at 0 (top panels) and 300 K (bottom panels). Doping leads to significant geometry change and atomic fluctuations, giving rise to strong electron−phonon coupling and fast quantum decoherence.

tion of time-dependent density functional theory (DFT)48,49 with nonadiabatic molecular dynamics (NAMD) theory.50,51 The FA0.75Cs0.25Pb0.5Sn0.5I3 structure is constructed by replacing 25% FA and 50% Pb with an identical concentration of Cs and Sn, respectively. The simulations show that the electron−hole recombination time scale of pristine FAPbI3 is on about 1 ns, agreeing well with experiment.52 Doping Cs and Sn into FAPbI3 leads to substantial geometry distortion and thus gives rise to a notable impact on electronic structures. First, doping narrows the band gap by 0.3 eV without creating trap states. Second, doping delocalizes hole wave functions significantly, which increases the overlap between the highest occupied (HOMO) and lowest unoccupied (LUMO) molecular orbitals, enhancing NA electron−phonon coupling. Third, the increased atomic fluctuations accelerate phononinduced loss of quantum coherence and strengthens NA coupling simultaneously. Overall, the rapid quantum decoherence competes successfully with the small band gap and large NA coupling, reducing charge recombination by a factor of 2.8 and extending the charge carrier lifetime to several nanoseconds. The obtained results indicate that rapid quantum decoherence, arising due to doping-induced pronounced nuclear dynamics, inhibits electron−hole recombination in Cs/Sn-doped FAPbI342 and suggests meaningful insights for the design of high-performance perovskite materials and devices. The NAMD simulations are performed with a decoherencecorrected surface hopping algorithm50,51 coupled with a classical path approximation implemented within time-dependent Kohn−Sham (KS) theory.48,49 The lighter and faster electrons are treated quantum mechanically, and the heavier and slower nuclei are treated semiclassically.49,53,54 Decoherence reflects the destruction of superpositions formed between a pair of electronic states via NA electron−phonon coupling, which drives nuclear trajectory branching.51 The decoherence time can be computed as the pure-dephasing time in optical response theory.55 In the present cases, decoherence needs to incorporate into the simulations because the decoherence time 3434

DOI: 10.1021/acs.jpclett.9b01330 J. Phys. Chem. Lett. 2019, 10, 3433−3439

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The Journal of Physical Chemistry Letters

with the same Cs/Sn doping concentration.42 In our NAMD simulations, we have scaled the calculated band gaps to the experimental values in order to directly mimic real environments. Typically, a smaller band gap tends to create large NA electron−phonon coupling because it makes the electronic and vibrational quanta closer to resonance. Besides the band gap, NA coupling plays an important role in driving nonradiative electron−hole recombination, which depends on both the nuclear velocity, dR/dt, and the overlap of the LUMO and HOMO state wave functions, ⟨ϕLUMO|∇R| ϕHOMO⟩. Table 1 shows that doping induces larger atomic motion than that in the pristine FAPbI3. Insets of Figure 2a,b show the charge densities of the HOMO and LUMO obtained from a representative MD snapshot. The insets demonstrate that doping brings increased charge delocalization on both the LUMO and HOMO, in particular, for the HOMO, as a result of increasing the mixing between the LUMO and HOMO compared with the pristine case. Both factors favor increasing the NA coupling in the doping case, Table 2. In order to

Besides the reported bond lengths, the canonically averaged standard deviations in the positions of each component (i) can provide additional information for nuclear dynamics. The data can be computed based on the formula σi = ⟨(ri − ⟨ri⟩)2 ⟩ , where ri stands for the component i along the 2 ps MD trajectories. Then, we average the standard deviations of all components in both systems. Table 1 shows that the standard Table 1. Standard Deviations (Å) in the Positions of the FA, Cs, Sn, Pb, and I Components for FAPbI3 and Cs/Sn-Doped FAPbI3 FAPbI3 doped

FA and Cs

Pb and Sn

I

0.223 0.245

0.216 0.253

0.207 0.248

deviations in the positions of component i decrease from the doped system to the pristine FAPbI3, consistent with the analysis obtained from bond lengths. The observation is reasonable because smaller Cs+ and Sn2+ substitution to larger FA+ and Pb2+ increases the lattice free volume and allows atoms to undergo substantial movements. Figure 2 presents the projected density of states (PDOS) of the pristine FAPbI3 and the Cs/Sn-doped FAPbI3. For the

Table 2. Experimental Band Gaps, Calculated Average Absolute NA Coupling, Pure-Dephasing Time, and Nonradiative Electron−Hole Recombination Time for FAPbI3 and Cs/Sn-Doped FAPbI3

FAPbI3 doped

band gap (eV)

NA coupling (meV)

pure-dephasing (fs)

recombination (ns)

1.5042 1.2242

0.44 0.69

6.81 4.25

1.60 4.40

provide much more solid proof of the arguments on the analysis of charge densities, we computed the inverse participation ratio (IPR)65 of the LUMO and HOMO along the 2 ps MD trajectories at 300 K (Figure 3), which

Figure 2. PDOS of (a) FAPbI3 and (b) Cs/Sn-doped FAPbI3 calculated using the optimized geometries. The PDOS of the pristine FAPbI3 is separated into contributions from FA, Pb, and I. Besides the above components, Cs and Sn orbitals are also composed of the Cs/ Sn-doped FAPbI3. The zero energy is set to the Fermi level. A comparison of insets between (a) and (b) indicates that doping increases charge delocalization on both the HOMO and LUMO, in particular, for the HOMO, enhancing their overlap and thus NA electron−phonon coupling.

Figure 3. IPR of LUMO and HOMO charge densities for FAPbI3 and Cs/Sn-doped FAPbI3 along the 2 ps MD trajectories.

pristine FAPbI3, the HOMO and LUMO are primarily supported by I and Pb atoms (Figure 2a). Besides I and Pb, Sn contributes to band edge states after Cs/Sn doping (Figure 2). Thus, I, Pb, and/or Sn generate the dominant NA electron−phonon coupling for electron−hole recombination. FA and/or Cs components do not contribute to the band edges but affect coupling via perturbing the inorganic framework. The NA electron−phonon coupling drives electron−hole recombination across a wide band gap of 1.29 eV. This calculated value is in agreement with the previous DFT calculations64 and is also close to the experimental value of 1.50 eV.42 After doping, the band gap decreases to 1.0 eV, showing good agreement with the reported 1.22 eV of FAPbI3

characterizes the extent of charge localization of KS electronic states. The IPR of a particular KS state is defined by IPR = N ·

∑ ki 4 2 2

(∑ k i ) IPR ∈ (0, 1]

where N denotes the number of grids of a given KS electronic state and ki represents charge density within the volume of a unit grid. The larger the IPR value, the more charge localized the state. An IPR of 1 indicates that charge is entirely localized 3435

DOI: 10.1021/acs.jpclett.9b01330 J. Phys. Chem. Lett. 2019, 10, 3433−3439

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The Journal of Physical Chemistry Letters

thermal fluctuation of the HOMO−LUMO energy gap, δEij(t), between electronic states i and j. Apparently, a large initial value of Cij(0) leads to rapid decoherence. In addition, the decoherence rate can be computed by either Truthlar’s decayof-mixing67,68 or Subtonik’s augmented SH approaches.69 However, the original Truthlar’s method suffers from a drawback that the decoherence rate depends on predefined parameters and limits its transferability for different systems. Recently, Prezhdo and coauthors have improved this algorithm in optical response theory.70 In contrast, estimation of a decoherence rate in Subtonik’s augmented approach has to compute the force differences between the occupied adiabatic state and all other states. It is not convenient to compute the decoherence time on-the-fly efficiently in electronic structure software. The pure-dephasing functions are shown in Figure 5, and the unnormalized ACF curves are presented in the inset.

to one site. The time evolution of the IPR of the LUMO for the Cs/Sn-doped FAPbI3 is similar to that in the pristine FAPbI3, approved further by the canonically averaged IPR value, 2.7940 × 10−5 vs 2.7694 × 10−5. In contrast, the evolution of the IPR of the HOMO for the doped system is consistently smaller than that in the pristine system, supported by the canonically averaged IPR, 2.0958 × 10−5 vs 3.1840 × 10−5. Therefore, the overlap between the HOMO and LUMO along the whole 2 ps MD trajectories of the Cs/Sn-doped FAPbI3 is larger than that in the pristine FAPbI3, leading to a strong NA electron−phonon coupling (Table 2). Electron−vibrational interactions produce both inelastic and elastic electron−phonon scattering. Inelastic scattering dissipates the electronic energy to phonons, while elastic scattering destroys the coherence of superpositions formed between the LUMO and HOMO via NA coupling. The inelastic scattering depends on the overlap of the HOMO and LUMO wave functions. Elastic scattering leads to decoherence, known as pure-dephasing in nonlinear optical response theory.55 In order to characterize the phonon modes that couple to electronic subsystems, we computed spectral density by performing Fourier transforms (FTs) of autocorrelation functions (ACFs) of the HOMO−LUMO energy gap fluctuations. Figure 4 shows that low-frequency vibrations

Figure 5. Pure-dephasing functions for the HOMO−LUMO transition in FAPbI3 and Cs/Sn-doped FAPbI3. The pure-dephasing times are obtained using Gaussians fitting, Table 2. The inset shows the unnormalized ACF. Typically, the greater the initial value, the faster the pure-dephasing.72

Fitting the data shown in Figure5 by a Gaussian function, exp[−0.5(t/τ)2], gives the pure-dephasing times (Table 2). The computed pure-dephasing time is 6.81 fs for the pristine FAPbI3 and 4.25 fs for the Cs/Sn-doped case. Under the cumulant approximation, the pure-dephasing time is determined by the decay of the function and the unnormalized ACF.71,72 Shown in the inset of Figure 5, both functions decay on a similar time scale and oscillate on almost identical periods in several hundred femtoseconds. In turn, the rate of puredephasing is governed by the initial value of the unnormalized ACF, which equals the square root of the electronic energy gap fluctuation. In general, a greater initial value of the unnormalized ACF facilitates rapid pure-dephasing.71,72 Apparently, the initial value of the Cs/Sn-doped FAPbI3 is larger than that in the pristine FAPbI3, leading to a shorter dephasing time. Quantum transition becomes slow if the dephasing time is short, which even stops at the dephasing time approaching infinitesimal, exemplified by the quantumZeno effect.73 Figure 6 presents the time evolution of the LUMO population in the FAPbI3 with and without Cs/Sn doping, characterizing electron−hole recombination dynamics. Fitting the data using the short-time linear approximation to the exponential decay P(t) = exp(−t/τ) ≈ 1 − t/τ, obtains the

Figure 4. FTs of ACFs for the fluctuations of the HOMO−LUMO gaps in (a) FAPbI3 and (b) Cs/Sn-doped FAPbI3. Lighter and faster Cs/Sn doping accelerates quantum decoherence.

dominate the entire spectrum in both systems and serve to create NA electron−phonon coupling, induce loss of coherence, and drive nonradiative electron−hole recombination. For pristine FAPbI3, the dominant peak at 33 cm−1 can be assigned to Pb−I bond bending and FA molecular librations.66 The peak at 101 cm−1 is associated with Pb−I bond stretching and FA molecular libration, while the peak at 167 cm−1 can be attributed to FA molecular torsion.66 Replacing larger and heavier FA/Pb with smaller and lighter Cs/Sn accelerates quantum decoherence and enhances NA coupling simultaneously due to the enhanced thermal fluctuations. Decoherence is another factor affecting electron−hole recombination besides the band gap and NA coupling, and its time can be computed as the pure-dephasing time in nonlinear optical response theory55 on the second-order cumulant approximation according to the equation Dij(t) =

(

exp −

1 ℏ2

t

t′

)

∫0 dt ′ ∫0 dt ″ Cij(t ″) . Here Cij(t) is defined as Cij(t)

= ⟨δEij(t′)δEij(t − t′)⟩t′. It is the unnormalized ACF of the 3436

DOI: 10.1021/acs.jpclett.9b01330 J. Phys. Chem. Lett. 2019, 10, 3433−3439

Letter

The Journal of Physical Chemistry Letters

thermal fluctuations that accelerate quantum decoherence despite increasing NA coupling simultaneously. Consequently, rapid decoherence competes successfully with a small band gap and strong NA couplings, suppressing electron−hole recombination and extending the charge carrier lifetime to several nanoseconds. The small NA coupling, sub-1 meV, and short decoherence time, sub-10 fs, favor a long-lived excited state and high PCE of mixed ion perovskite solar cells.42 The simulation emphasizes the key role of quantum coherence in affecting photoexcitation dynamics in condensed-phase perovskite materials, which has been emphasized in biological systems for a long time,74 and provides meaningful guidance for improvement of halide perovskite optoelectronic and photovoltaic devices.

■

Figure 6. Time evolution of the LUMO population for FAPbI3 and Cs/Sn-doped FAPbI3.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

nonradiative decay times, shown in Table 2. The relaxation time is 1.60 ns for the pristine FAPbI3 and 4.40 ns for the Cs/ Sn-doped FAPbI3, showing excellent agreement with experiment.42 The extension in the charge recombination time after doping can be rationalized by the band gap, NA coupling, and pure-dephasing time. In the present study, the smaller band gap and larger NA coupling in Cs/Sn-doped FAPbI3 compared to those in FAPbI 3 accelerate charge recombination. Importantly, the shorter decoherence time of the doped system compared to that in the pristine system slows charge recombination. Overall, the rapid decoherence competes successfully with the small band gap and strong NA coupling, inhibiting electron−hole recombination. A tens of nanoseconds charge recombination time scale is long and originates from both the small NA coupling (