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Nonradiative Relaxation in Real-Time Electronic Dynamics OSCF2: Organolead Triiodide Perovskite Triet S. Nguyen and John Parkhill* Department of Chemistry and Biochemistry, The University of Notre Dame du Lac, 251 Nieuwland Science Hall, Notre Dame, Indiana 46556, United States S Supporting Information *

ABSTRACT: We apply our recently developed nonequilibrium real-time time-dependent density functional theory (OSCF2) to investigate the transient spectrum and relaxation dynamics of the tetragonal structure of methylammonium lead triiodide perovskite (MAPbI3). We obtain an estimate of the interband relaxation kinetics and identify multiple ultrafast cooling channels for hot electrons and hot holes that largely corroborate the dual valence− dual conduction model. The computed relaxation rates and absorption spectra are in good agreement with the existing experimental data. We present the first ab initio simulations of the perovskite transient absorption (TA) spectrum, substantiating the assignment of induced bleaches and absorptions including a Paulibleach signal. This paper validates both OSCF2 as a good qualitative model of electronic dynamics, and the dominant interpretation of the TA spectrum of this material.

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INTRODUCTION Rational design of functional materials requires a strong understanding of the characteristic dynamics of electronic excited states. However, simulating these processes is challenging since they involve time scales separated by many orders of magnitude. Real-time time-dependent density functional theory (RT-TDDFT) is a powerful yet affordable ab initio technique for modeling subnanosecond electronic dynamics in molecular systems.1 It propagates the electronic density of the system-of-interest in real time but lacks any treatment of energy dissipation. Recently, we developed timedependent open self-consistent field at second order (OSCF2), a black-box RT-TDDFT method that includes dissipation and naturally respects detailed balance.2−4 The method assumes no model Hamiltonian, and we have shown that it can be used to qualitatively reproduce several features of electronic dynamics in insulators and semiconductors. Most recently, we generalized the method to calculate transient absorption (TA) spectra, with only the molecular structure of the system as input.4 Despite these successes, we would like to continue validating our approach to systematically understand the strengths and weaknesses of our approximations. This paper applies the technique to lead triiodide perovskites, a particularly appropriate target for a theory of phonon-mediated electronic relaxation. Within the last 6 years, the power conversion efficiencies of perovskite-based solar cells have been rapidly driven to more than 20%,5−17 making them the front runner of next-generation photovoltaics. The unprecedented progress of this class of materials come from its unique combination of attractive © XXXX American Chemical Society

properties: tunable band gap, ambipolar transport properties, long charge-carrier diffusion length, and solution processability.18−22 They have also been studied intensively for other applications including in light-emitting diodes, optically pumped lasers, and water-splitting assemblies.23−26 In this class, methylammonium lead triiodide perovskite (MAPbI3) is the most well-characterized system.27,28 Rapid experimental advances motivate further theoretical understanding of the underlying photophysical processes. The stationary properties of the ground and excitonic states of this material have been thoroughly studied.29−35 It is known that Kohn−Sham generalized gradient approximation (GGA) calculations yield qualitatively correct band dispersion and wave functions, and fortuitously deliver the optical gap within 0.1 eV of the experimental value.30,36,37 As usual, this success of density functional theory (DFT) rests on a cancellation of errors, in this case between many-body exchange and spin−orbit effects. Several research groups have recently reported DFT-GW calculations that reproduce the experimental optical gap38,39 and the absorption spectrum of the tetragonal phase,40 which is relevant to the operational temperature range of solar cell applications. However, our understanding of the relaxation pathways in perovskites leaves more to be desired. The significant cost of these calculations requires us to use DFT as our electronic structure model.41 Received: July 11, 2016 Revised: August 9, 2016

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DOI: 10.1021/acs.jpca.6b06937 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

consequently, decay rates in these methods do not relax toward the correct equilibrium distribution of populations at long times. The electronic dynamics can produce the linear absorption spectra by Fourier-transforming the resulting dipole oscillations μ after applying an impulsive field to the system:45 4πω σ(ω) = ∑ Im{μi (ω)/Ei(ω)} 3c i = x , y , z (3)

Because of its black-box nature, OSCF2 allows us to perform simulations which mirror spectroscopic measurements. In this paper, we will investigate real-time ab initio simulations of relaxation processes in tetragonal MAPbI3 to substantiate the interpretation of transient spectra developed in recent experiments.6,11,12,22,42−44 We employ OSCF2 to make quantitative, atomistic models of the excited states and simulate the subpicosecond dynamics of relaxation channels.45,46 Particularly, we will show that our dynamics simulations corroborate a phonon-mediated relaxation between what has been called a dual-valence band or a charge-transfer band47 to explain the photoinduced bleach signals at 480 and 760 nm. We also aim to justify OSCF2 as an affordable and useful tool to study ultrafast electronic dynamics.

Compared to the stick spectra produced by adiabatic RTTDDFT, linear-response (LR) spectra made by OSCF2 are naturally broadened as a result of relaxation in our theory, and can be directly correlated to experimental spectra. This method can also simulate transient spectra.4,50 In a TA simulation, the system is propagated three times during sampling time tabs: (1) propagation Pe(t) which starts from the pumped density and experiences a probe pulse, (2) a pumped correction Pref(t) which receives no probe, and (3) a ground-state propagation Pgs(t) which only receives the probe. The resulting dipoles are subtracted, and the TA spectrum for each pump−probe delay time is simply:

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THEORETICAL METHODS Our results are drawn from OSCF2 as implemented in the QChem quantum chemistry package48 which we will briefly summarize. We refer the readers to refs 2 and 3 for the full description of the implementation. Most significantly, OSCF2 allows us to simulate subnanosecond photoinduced electronic dynamics without assuming any basis of stationary states. This is an important feature for systems like perovskite which rapidly thermalize between closely related excitonic and free-carrier states. The method extends the equation-of-motion of ordinary RT-TDDFT for the one-electron density matrix, γ, with a dissipative correction derived from second-order perturbation theory. The Liouville-like equation of TDDFT with Markovian dissipative correction has the following form: γ̇ =

−i ̂ 1 [F {γ }, γ ] − 2 9{γ } ℏ ℏ

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(4)

RESULTS AND DISCUSSIONS We focus on the tetragonal structure which is the stable phase of MAPbI3 perovskite at room temperature51−54 (Figure 1a).

(1)

Here nonorthogonal bases of atom-centered functions (the AO basis) are labeled by Greek characters (|μ⟩,|ν⟩...), and the orthogonal basis of one-electron orbitals are labeled (ϕa(r),ϕb(r)...). The one-electron density matrix is the expectation value of a one-particle operator γab = ⟨aa ab⟩, where aa and ab are Fermion creation and annihilation operators.49 Each atomic basis function experiences a unique dissipative environment 9{γ }ba that takes the form

Figure 1. (a) Crystal structure of MAPbI3 perovskite in the tetragonal phase (Pb, gray; I, purple; MA groups not shown). (b) Embedding scheme in which a cluster is surrounded by roughly 2 nm of crystal, represented as point charges in all directions.

9{γ }ba = {Sa , f , f , c Γν(ωcf )Vaeν V νfcγbcηfe + Sb , f , f , d (Γν(ωdf )Vbeν V νfd)†

The crystal structure shows a PbI6 octahedral distortion along the c-axis. From the optimized periodic structure, we constructed a cluster model with molecular formula (CH3NH3)2Pb8I36 for OSCF2 simulations. To mimic the effects of crystalline structure, the cluster is embedded in an effective electrostatic field following Carter and co-workers55 (Figure 1b). Embedded cluster models are a common and validated approach often combined with costly electronic structure calculations. This technique is reasonably justified as the size of the cluster in this work is comparable to that of the characteristic photoexcited states56,57 of the material. The embedding procedure is described in Supporting Information. In what follows, we will validate the physicality of the chargeembedded model by comparing the computed absorption spectra with experiments and periodic calculations. Spectral Density. The frequency-dependent system−bath coupling made for each atomic orbital, the spectral density (SD), is one of the most important quantities in an OSCF2 calculation. It quantifies and chemically characterizes the motions responsible for energy dissipation. We compute the

ν † γdaηfe − Sd , b , a , c Γν(ωca)Vdeν Vacν γdcηbe − Sc , a , b , d (Γν(ωdb)VceνVbd )

γdcηea}

ΔA(ω) ∝ Im(-{Pe(t ) − Pgs(t ) − Pref (t )}/Eprobe(ω))

(2)

where Sa,b,c,d = δabδcd + δbcδad(1 − δabδcd) The dissipative term 9 depends on the product of molecular orbital coefficients Vi,jν = ∑ν Ci,νC†j,ν, the one-hole density matrix ηpq = δpq − γpq, and a matrix of Markovian transition rates for each atomic function evaluated at the frequencies of orbital energy differences Γν(ωab), where ℏωab = ϵa − ϵb. The Markovian rates can be computed from the spectral density that quantifies the atom-centered system-bath couplings. We employ eq 1 to simulate electronic dynamics of the system by applying simulated laser fields. In contrast to existing nonadiabatic dynamics methods, this work examines two important features of the system. It introduces an atomistic model of energy loss to vibrations of the crystalline environment, and it thermalizes self-consistently to a Fermi−Dirac distribution. Typical RTTDDFT methods only predict Pauli blocking for light absorption processes but not for nonradiative relaxation; B


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The Journal of Physical Chemistry A SD from the atomic-orbital energy fluctuations along an equilibrium ab initio molecular dynamics (AIMD) trajectory. Since most energetic fluctuations are caused by phonons in the material, the SD summarizes bond fluctuations with strong electronic coupling (see Supporting Information). Besides the structural fluctuations, methylammonium (MA) lead halides are known to be dynamically disordered due to fluctuations of the MA moiety31,51 (inset of Figure 2). Multiple

the consequences of MA vibrational modes in phonon relaxation in the following section. Real-time electronic dynamics. We simulate electronic dynamics of tetragonal perovskite by propagating eq 1. The initial density is the ground-state density of a routine SCF calculation. The system is excited by a minimum-uncertainty electric field pulse polarized along the transition dipole moment of the lowest exciton, generating a superposition of bright excited states. As the simulation progresses, the one-electron density γ(t) oscillates and relaxes to thermal equilibrium, providing the time-dependent dipole moment μ(t), electron and hole densities, and other properties of interest. Over 600 fs of dynamics thermalization were obtained. All DFT calculations, unless specified, were performed using the B3LYP functional with the LANL2DZ basis set (total 1304 basis functions), with additional polarization functions and relativistic effective core potentials for Pb and I atoms.62 Propagation of a large system like this at the ab initio level is costly: 1300 wallhours per picosecond of simulation time on a 64-core 2.4 GHz node, and would be intractable with the addition of spin−orbit coupling (SOC).63 Details of dynamics parameters are presented in Supporting Information. The absorption spectrum of our cluster model measures the realism of OSCF2 theory. Figure 3a shows the computed absorption spectrum averaged over geometries from equilibrium AIMD; it is homogeneously broadened because of relaxation. The agreement obtained with existing experimental spectra for tetragonal MAPbI3 is good considering the relatively inexpensive model chemistry employed. OSCF2 spectrum mirrors the GW/Bethe-Selpeter equation (GW/BSE) spectrum calculated by Zhu et al.38 The first exciton peak occurs at 1.70 eV, slightly blueshifted relative to its experimental counterpart (1.60 eV27) and the best computed values to date (1.5639 and 1.57 eV38). The second peak at 2.1 eV red-shifted from the GW/BSE spectrum is due to larger energy gaps seen in the upper conduction band when spin−orbit coupling is included.63 DFT-based band structure calculations reported the energy of the lowest exciton transition within 0.1 eV of the experimental value without taking into account SOC of the heavy atoms.29,57 Moreover, the 0.1 eV discrepancy in energy of the transition is typical with the best model chemistry we can propagate.64 The rates of nonadiabatic relaxation we obtained from OSCF2 simulations are the key results of our paper. By performing a short-time Fourier transform (STFT) of the

Figure 2. Spectral densities for a 6p orbital on a Pb atom and a 5p orbital on an I atom. Inset: Histogram of the angular deviations of the MA group from its equilibrium orientation in AIMD simulation.

studies have shown that dynamics of MA can alter the optical properties of this class of materials.58 The MA rotational motion in the inorganic cage has been found to be very fast (4− 6 ps).51,59,60 As expected, the lighter MA moiety dominates the nuclear dynamics of AIMD trajectory and features of the SD Figure 2 shows the SD of two atomic orbitals on Pb and I atoms. Since the I 5p orbitals participate significantly in the hole state and the Pb 6p orbitals predominate the electron state, these SDs largely quantify the importance of vibrational modes that mediate relaxation in the material. The plot clearly shows that MA groups strongly modulates the energy of an electron which would occupy the hole and the electron states. The low-frequency region of this plot resembles the experimental Raman spectrum reported by Quarti et al.:61 intense features below 300 cm−1 are associated with MA vibrations, although note that these two quantities are only indirectly related. The NH3+ and CH3 stretching modes in the 3000 cm−1 region also play an important role. We will discuss

Figure 3. (a) Absorption spectrum of MAPbI3 perovskite computed with OSCF2 (thick line) compared with GW/BSE spectrum by Zhu et al. (digitized from data in ref 38, thin line). The OSCF2 spectrum has been scaled to match the intensity of GW/BSE spectrum in correspondence of the lowest exciton transition. (b) Spectrogram showing the time evolution of excited states. C


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hot-electron cooling (channels within the conduction band) contributions. Intraband relaxation has been observed in multiple transient spectroscopy experiments: hot-hole cooling reported by Xing et al.67 and Yamada et al.68 and hot-electron cooling reported by Yang et al.,47 all with subpicosecond lifetimes. Hot electrons appear to cool down to the band edge faster than hot holes, consistent with experimental findings. The greater disorder of the tetragonal structure leads to more fluctuations and a denser manifold of states, both of which contribute to faster relaxation. These OSCF2-GR rates highlight phonon-mediated relaxation, but miss the dynamic Pauli-blocking which plays an important role in the photophysics of this material.74 We extracted the corresponding lifetimes from OSCF2 dynamics by fitting regions of real-time spectrogram plots to exponential functions. Overall, time scales extracted from OSCF2 dynamics are in quantitative agreement with the GR estimations, and match the characteristics of carrier thermalization measured in the material.47,67,68 However, it should be noted that OSCF2 relaxation rates are not constant due to Pauli-blocking factors included in our theory. We have previously shown that nonradiative relaxation from an excited state undergoes a period of transient acceleration before entering a slower blocked, essentially exponential rate.3 As a consequence, the GR time scales predicted by eq 5 are usually a lower limit of those extracted from OSCF2 dynamics. Shown in Figure 4, the subpicosecond intraband relaxation is well

dipole as it relaxes back to its stationary value during the course of propagation, the decay of excited-state populations can be measured in a fashion very close to experiment. Because we do not have to make any type of tight-binding approximation, we do not need to make an ambiguous link between state populations and dipole signal. At each time, we can correlate the dipole signal to the band populations of the cluster to separate the effects of population relaxation and the transition dipole moment. The STFT dipole spectrum is called a spectrogram; it is shown in Figure 3b. There are essentially two regions heavily populated after the impulse excitation: the sharp gap-exciton band centered at 1.70 eV, and a large diffuse population of carrier in the conduction band region above 2 eV. The low-energy band corresponds to the lowest excitonic transition and has single-exponential decay character. The diffuse band has two visibly distinct regimes, one rapid relaxation and one slower relaxation, both with subpicosecond time scales. The spectrogram is in some ways easier to interpret, because it has no contribution from the ground state of the system. However, it lacks the direct connection with experiment offered by the transient spectrum which will be discussed later. The decay of the dipole mixes nearly degenerate transitions like time-resolved spectroscopy. To further clarify the electronic states involved in the relaxation channels, we computed golden rule (GR) rates which are specific to individual transitions:65 kc ← a = 2Γac ,ca(ωac) =

1 V caS|2 ℏ2

B B eiωacτ ⟨δVÎ (τ )δVÎ (0)⟩

∞

∫−∞ dτ (5)

where = ⟨c V̂ S a⟩ is the coupling matrix element between the relevant levels, and C(τ) = ⟨δV̂ IB(τ)δV̂ IB(0)⟩ is the bath correlation function calculated from the SD. The decay time scales of channels near the gap provide qualitative information about excitonic and carrier states in the material. The lowest excitonic transition has more than 95% band-to-band character, so relaxation of this state is essentially the interband relaxation rate. This time scale is on the order of nanoseconds, multiple orders of magnitude longer than other relaxation channels. Our computed lifetime lies in the low end of experimental reports for solution-processed polycrystalline perovskite films (4.5−140 ns).22,43,66−68 Our models capture phonon- and polarizationmediated relaxation which experiments have suggested are dominant in these materials. The Markovian approximation, which tends to predict overdamp compared with nonMarkovian equations-of-motion,69 contributes to the significant overestimation of OSCF2 rates. Hirasawa et al.56 attribute the photoexcited states in MAPbI3 perovskites to Mott−Wannier excitons with large Bohr radii (2257−2856Å). It is widely accepted that the exciton binding energy in this material is very low (≈ 50 meV),44,56,57,70−72 significantly smaller than organic semiconductors. Free charge carriers are produced in the material shortly after light absorption. Spectroscopic experiments measure the time scale of a nongeminate recombination process;42,68 the mobility of these carriers in MAPbI3 perovskite is remarkably high compared with other solution-processed materials. 73 Lifetimes are also sensitive to the defect concentration of the sample, which is difficult to consistently control. The GR predicts a large number of accessible subpicosecond relaxation channels. We separated the thermalization processes into hot-hole cooling (channels within the valence band) and VSca

Figure 4. Time evolution of a hot-electron cooling channel (solid line) extracted from the spectrogram in Figure 3, and the exponential fit (dashed line) which gives it a time scale of 128 fs. Relaxation of the electron density is evident from the density plots at the beginning and at the end of the propagation.

captured in the length of our propagation time; this region can be reasonably fitted to a biexponential decay. Our dynamical simulations cannot access long enough time and length scales to study complete electron−hole recombination, but fast cooling channels for hot holes and hot electrons are captured nicely by this procedure. The computed time scales from the GR and the fitted dynamics of MAPbI3 perovskite are summarized in Table 1. We simulated the zero-delay TA spectrum of the system to further validate our method. This was done by performing a propagation starting from density of the first excited state and subtracting the resulting dipole oscillation from the ground state.4 Shown in Figure 5, the simulated TA spectrum consists of three regions: two negative bands at 450 nm and at 700 nm, and a broad positive band between 480 and 640 nm, which largely reproduces the experimental counterpart from Manser D


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SOC brings about significant changes to the band structure, SOC mainly perturbs the conduction bands, leaving the valence bands and the spacing between valence bands involved in TA unaffected. The narrowing of the gap brought about by SOC is reversed by the electron−hole interaction in GW, so the interband relaxation rates would also be very similar. Splitting in conduction bands seen in the SOC-DFT band structure suggests that hot electrons would undergo faster relaxation in OSCF2 if SOC interactions were incorporated. Finally, we discuss the contribution of MA vibrational modes in phonon relaxations of this material. It is evident from the strength of the MA vibrational markers in Figure 2 that the organic cations significantly mediate relaxation in a key excited state. We measured the contribution of the dominant vibrational frequencies, NH+3 stretching modes, by computing OSCF2−golden rule time scales using the SD that takes only these modes into account. In this scenario, we obtained a 70% increase in the interband relaxation time scale, and hot carriers would cool in hundreds of picoseconds. This suggests that there is a connection between the symmetry (or lack thereof) of the inorganic cage and the organic cation’s dynamics in relaxation channels of perovskite. Hence as the temperature increases, the phase transition from the tetragonal structure to the more ordered pseudocubic structure could potentially hinder the effectiveness of perovskite-based solar cells, as recently pointed out by Quarti et al.81

Table 1. Computed and Experimental Relaxation Time Scales for Hot Holes, Hot Electrons, and Interband Transitions of MAPbI3 Perovskitea process experimental OSCF2− golden rule OSCF2

hot-hole cooling (fs)

hot-electron cooling (fs)

interband relaxation (ns)

300,68 40067 324−635

3047 95−549

4.5−14022,43,66−68 5.1

620

128

N/A

a

We report the range of time scales for a process with multiple channels.

Figure 5. Transient absorption spectrum of tetragonal MAPbI3 perovskite computed with OSCF2 (delay time τ = 0, thick line) compared with its experimental counterpart from Manser et al. (τ = 5 ps, digitized from data from ref 42, thin line). The OSCF2 spectrum has been scaled to match the maximum intensity of experimental spectrum. Schematic band diagrams with dual valence−dual conduction model correspond to regions of the spectrum are also shown.

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CONCLUSIONS

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ASSOCIATED CONTENT

In summary, we have presented real-time OSCF2 simulations of light absorption in the tetragonal structure of MAPbI3 perovskite. We reproduced the interband relaxation time scale and identified multiple ultrafast cooling channels for hot electrons and hot holes. OSCF2 dynamics substantiates a dual valence-dual conduction, phonon-mediated relaxation model of the transient spectra in this material and provided a detailed picture of many subpicosecond relaxation channels observed in ultrafast experiments. The higher-energy bleach in the transient spectrum was shown to be a physical consequence of Pauliblocking, which is included in our theory. Our findings demonstrated that the dynamics of the organic cation has important consequences in relaxation, and in the tetragonal-topseudocubic transition. Since MAPbI3 perovskite bears many similarities in electronic structure with other hybrid organic− inorganic perovskites, these simulations can potentially open the door to visualizing the thermal charge separation process in this class of materials, in real time. We view this as a significant opportunity to understand the interplay between thermalization and screening in gapped systems.

et al.42 The reduced intensity of the 700 nm negative signal is likely due to a stimulated emission component which is underestimated by TDDFT75 because of difficulties with resonant transitions.76−78 The lower-energy bleach corresponds to the band-edge transition (760 nm from ref 42.). The higherenergy negative band has a substantial overlap with the experimental 480 nm signal, and we found that this corresponds to excitations from states below the Fermi level. Our spectrum supports the dual valence band, first proposed by Xing et al.,42,67,75 which attributes the observed bleach signals to transitions from two valence bands to the conduction band minimum (VB1 → CB1 and VB2 → CB1). The higher-energy bleach also nicely demonstrates the Pauli-blocking effect that is captured by OSCF2. Since the system starts at the gap excitation, then excitation into the bottom of the CB is forbidden by spin statistics, resulting in the bleach. There is evidence that stimulated emission contributes to the 480 nm negative signal as well.75,79 The broad positive region matches the relative energies of hot-hole cooling channels in the valence band. Hot-electron cooling transitions correspond to longer wavelengths, and so these transitions are not observed in the TA spectrum. Both linear absorption and TA simulations support a dual valence band-dual conduction band model, first proposed by Marchioro.75 Spin−orbit coupling plays a role in band structure of MAPbI3.63,80 Treating it in our dynamics would more than quadruple the computational expense, making our simulations impossible. The DFT band structure of MAPbI3 with and without SOC reported by Even et al.63 shows that although

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.6b06937. Band structure calculations and optimized lattice parameters for tetragonal MAPbI3, description of the cluster model and embedding procedure, and description of the spectral density and parameters for electronic dynamics simulations in OSCF2 (PDF) E


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AUTHOR INFORMATION

Corresponding Author

*(J.P.) E-mail: [email protected]. Telephone: +1 (574) 631-2696. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the University of Notre Dame’s College of Science and Department of Chemistry and Biochemistry, NDEnergy and the Honeywell Corporation for generous start-up funding. T.S.N. gratefully acknowledges the support of the NSF Graduate Research Fellowship under Grant DGE-1313583. We thank John Herr for valuable comments.

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