Hybrid organic-inorganic perovskites (HOIPs) have received intense attention in photovoltaic and optoelectronic devices due to their excellent electronic and optical properties, including small effective masses of charge carriers [1, 2], high defect tolerance [3], long carrier lifetime and diffusion length [4, 5] and high optical absorption cross-sections [6]. These advantages enable HOIPs to be applied in a broad range of areas, such as lasers [7], photodetectors [8], light-emitting devices [9], transistors [10], photocatalytic water-splitting assemblies [11], and solar cells [12]. In particular, utilization as solar cells has made great success because the photovolatic conversion efficiency (PCE) of triple-organic-cation MAPbI₃ (MA = CH₃NH₃$ ^+ $) based cell has reached 25.2% [13] within just a decade, exceeding the record PCE of polycrystalline silicon solar cell and showing appealing commercial prospect. It is believed that the long-lived excited-state plays a benign role in promoting the solar cell performance because it reduces non-radiative charge and energy losses. The reduced recombination is associated with either large polarons formation due to rotation of MA cations or forming ferroelectroic domains owing to ordered alignment of MA cations. Both of them are able to reduce charge recombination by localizing and separating charge carriers. Our previous theoretical studies have rationalized the two experimental observations [14, 15]. However, the instability of MAPbI₃ exposure to humidity and light and heat irradiation constitutes a major issue for device applications [16]. The relatively large bandgap of 1.61 eV [17] for MAPbI₃ in tetragonal phase constitutes an additional obstacle for limiting its PCE approaching to the Shockley-Queisser limit for single junction solar cells [18].

Compared to MAPbI₃, FAPbI₃ (FA = CH[NH₂]₂$ ^+ $) has smaller bandgap (1.47 eV) [19], higher stability [20] and better carrier transport property [21], leading to the PCE of FAPbI₃ perovskite solar cells being 22.1% with a very high open-circuit voltage [22], which is higher than the record PCE of single-organic-cation MAPbI₃ solar cell [23] and is associated with long-lived charge carriers in FAPbI₃. Using femtosecond transient absorption spectroscopy and subpicosecond time-resolved terahertz spectrophotometry, Piatkowski et al. have reported the excited-state lifetime falls into the window between 700 and 1400 ps, which is about 200-300 ps in MAPbI₃ polycrystalline films [24]. Another experiment reported an extremely long charge carriers lifetime up to 1105 ns in a single crystalline FAPbI₃ by incorporating excess iodide ions into the precursor solution [22]. Our time-domain simulations demonstrated that a lead vacancy creates a shallow hole trap state that reduces electron-phonon coupling between the initial and final states for non-radiative electron-hole recombination, extending excited-state lifetime [25]. In addition to the extrinsic factors, intrinsic factors, such as rotation of FA cations, should also affect the charge carriers lifetime significantly, which was demonstrated in MAPbI₃ [15]. However, rotation of FA cation inside FAPbI₃ is believed to hardly happen due to its large ionic size. Experiments defy this exception. Seshadri and coauthors reported FA and MA exhibit very similar rotation rates (8 ps) in FAPbI₃ and MAPbI₃ at room temperature [26]. Shortly after, Oliver and coauthors observed that FA underwent 90$ ^{\circ} $ flips on picosecond time scales [27]. One may wonder how the rotation of FA cations affects the nonradiative electron-hole recombination rate in FAPbI₃. In order to answer this question, it is required to make an atomistic time-domain simulation of electron-hole recombination in FAPbI₃ by considering the rotation of FA cations. A detailed understanding of the underlying mechanism will provide valuable insights for design of high performance perovskite solar cells.

Motivated by recent experiments [22, 24, 26, 27], we have investigated the influence of rotation of FA cations on the nonradiative electron-hole recombination in FAPbI₃, using time-dependent density functional theory (TD-DFT) [28] combined with nonadiabatic molecular dynamics (NA-MD) [29-31]. The simulations show that reorientation of FA cations notably delays the recombination because it reduces NA coupling with respect to the pristine system. Such reorientation of FA cations only brings small change in bandgap that has little influence on the difference of the electron-hole recombination time scales for the two systems. Driven by the Pb-I bending and stretching modes, electron-hole recombination occurs within sub-nanosecond in pristine FAPbI₃, agreeing well with experiment [24]. By distorting the inorganic Pb-I lattice of the rotated FA cations, electron and hole wave functions are localized on the side and middle parts of the simulation cells, leading to the NA coupling decreasing by a factor of 2 compared to the pristine system. At the same time, the atomic motions are suppressed and decoherence time is prolonged. Consequently, the reduced NA coupling competes successfully to the increased decoherence time, extending the electron-hole recombination time scale to 3.17 ns. The retarded electron-hole recombination minimizes nonradiative charge and energy losses and is beneficial to optimizing the performance of perovskite solar cells.

The NA-MD simulations are carried out using the decoherence-induced surface hopping (DISH) method [32] implemented within the time-dependent Kohn-Sham framework [33]. The lighter and faster electrons are treated quantum mechanically, while the heavier and slower nuclei are described semi-classically. DISH algorithm treats nuclear wave functions branching and quantum decoherence correction on equal footing in the mixed quantum-classical approach. Decoherence, known as pure-depahsing time optical response theory [34], is incorporated into the NA-MD approach for simulating electron-hole recombination, because the decoherence time is several orders of magnitude shorter than the electron-hole recombination time [24]. The approach has been applied on studying photoexcitation dynamics in a broad range of systems [35-43], including hybrid organic-inorganic perovskite [35-40], all-inorganic perovskite quantum dots [41] and transition-metal dichalcogenides junctions [42, 43]. A detailed theoretical method can be found in Refs.[30, 31].

The geometry optimization, electronic structures, adiabatic MD, and NA coupling were calculated using the Vienna ab initio simulation package (VASP) [44]. The projector-augmented wave method was applied to describe the interactions between ionic cores and valence electrons [45]. The Perdew-Burke-Ernzerhof (PBE) functional was used to treat the electron exchange-correlation interactions [46]. The cutoff energy of the plane wave basis was set to 400 eV. The Grimme DFT-D3 approach was used to describe the van der Waals interactions [47]. The geometry optimizations were performed using 1$ \times $2$ \times $2 Monkhorst-Pack $ k $-point mesh [48]. A much denser $ k $-point mesh of 2$ \times $4$ \times $4 was employed to obtain the density of states [48]. The adiabatic MD and NA couplings were calculated at the $ \Gamma $-point because both the systems had direct bandgap at the $ \Gamma $-point. After geometry optimization at 0 K, two systems were heated to 300 K by repeated velocity rescaling. Then, 6 ps adiabatic MD trajectories were obtained in the microcanonical ensemble with 1 fs time step. Finally, all of the 6000 geometries were selected as initial conditions to simulate the electron-hole recombination using the PYXAID code [30, 31].

Spin-orbit coupling (SOC) is important in FAPbI₃ due to the presence of heavy Pb and I atoms. The PBE functional introduces the electron self-interaction error, and more rigious theory such as hybrid functional or the GW theory are required to cancel the error. The GW theory or a hybrid functional significantly overestimates the bandgap in lead halide perovskites, which should combine with the SOC for generating good bandgaps [49]. However, NA-MD simulation for a periodic system with GW or a hybrid functional combined with SOC is a computationally consuming task. In contrast, PBE functional is able to generate good bandgap values in lead halide perovskites [49], due to cancellation of the self-interaction error. An accurate energy gap is very important for simulating electron-hole recombination since it occurs across the fundamental bandgap in the present systems. The PBE functional has obtained good results on charge dynamics in lead halide perovskites that are comparable with experiments, including perovskites containing boundaries [50, 51] and lead vacancy [25], cation mixing [52] and forming localized charge [15].

The optimized lattice constant of $ \alpha $-phase FAPbI₃ is 6.41 Å, agreeing well with the experimental value of 6.36 Å [53]. By expanding the optimized unit cell to a (4$ \times $2$ \times $2) supercell, the 192-atom system forms to represent pristine FAPbI₃ (FIG. 1(a)). Then, rotating half of FA cations (90$ ^{\circ} $) perpendicular to the N-N axis in pristine FAPbI₃ leads to the R-FA structure (FIG. 1(b)). FIG. 1(a) shows that all the FA cations reside orderedly within the inorganic octahedra and the octahedra themselves remain perfect as well, due to that FAPbI₃ forms an ideal perovskite structure. Rotating partial FA cation alters not only the H$ \cdots $I hydrogen bond length but also the hydrogen bonding direction, causing the distinct distortions of the Pb-I lattice around the rotated FA cations, which directly influence the electron-phonon coupling [27, 49].

In order to quantitatively explore the effect of the rotation of FA cations on the atomic fluctuations, we compute the the standard deviation of the atomic positions based on the following formula:

$ \begin{eqnarray} \sigma_i = \sqrt{\langle({\vec{r_i} } -\langle \vec{r_i}\rangle)^2\rangle} \end{eqnarray} $  (1)

Here $ \vec{r_i} $ represents the location of atom $ i $ at time $ t $, and the angular bracket represents ensemble averaging. The calculated standard deviations for total, FA, Pb and I atoms are summarized in Table Ⅰ. The data show that the values of each component and overall system in the R-FA systems are smaller than that in the pristine system, indicating that the rotation of FA cations collectively suppresses the motions of themselves and the inorganic I and Pb atoms. The inhibited atomic motions weaken the electron-vibrational interacting, leading to NA coupling decreasing and phonon-induced loss of coherence increasing simultaneously. The two factors compete with each other and have an opposite influence on the quantum transition dynamics.

FIG. 1 (c) and (d) shows the projected density of states (PDOS) of the pristine and R-FA systems. The PDOS are split into contributions to FA, Pb and I components. Similar to previous studies [25, 49, 54], the I and Pb orbitals are primarily composed of the highest occupied (HOMO) and lowest unoccupied molecular orbital (LUMO), respectively. These atoms contribute to creating majority of NA coupling and leading to nonradiative electron-hole recombination across the fundamental bandgap. The FA has contribution neither to the HOMO nor to the LUMO. They affect the NA coupling in an indirect way. However, the electric field created by FA cations is much weaker than that created by MA cations because its dipole moment is about less than half of MA cation [55]. Therefore, the influence of FA cations on the inorganic Pb-I lattice via electrostatic interaction is expected to be small. In turn, they affect the inorganic lattice and NA coupling due to the large size of FA cations. The HOMO and LUMO are separated by a wide bandgap of 1.47 eV for the pristine FAPbI₃, agreeing with experiment [19]. Rotation of partial FA cations slightly increases the bandgap to 1.50 eV. The small difference in bandgap has little influence on electron-hole recombination. Consequently, NA coupling and loss of coherence significantly affect the excited-state lifetime.

The strength of NA coupling depends on the overlap of the LUMO and HOMO charge densities, $ -i\hbar\langle \phi_j $$ |\nabla_R|\phi_k \rangle $, as well as the nuclear velocity, d$ \bm R $/d$ t $. A larger mixing of the corresponding charge densities and faster atomic motions usually generate a stronger NA coupling and a faster charge recombination. FIG. 2 demonstrates the charge densities of HOMO and LUMO in the two systems. They are consistent with the analysis of PDOS. The HOMO and LUMO of pristine FAPbI₃ are primarily supported by the I and Pb atoms (FIG. 2(a)). Rotating FA cations lead to the HOMO prone to localizing on the distorted inorganic sub-lattice while the LUMO localizing on the distant region of the distorted part (FIG. 2(b)), decreasing the overlap of the two wave functions and the NA coupling. The reduced atomic motions in the R-FA system, shown in Table Ⅰ, reduces the NA coupling further, leading to the NA coupling being half of that in the pristine system, 0.55 meV versus 1.10 meV (Table Ⅱ).

In order to identity the phonon modes participating in the nonradiative electron-hole recombination, we computed the spectral densities (Eq.(3)) by performing Fourier transform of the autocorrelation function (Eq.(4)) of the bandgap fluctuation for the pristine and R-FA systems.

$ \begin{eqnarray} I(\omega)& = &\left|\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \rm{d}t \rm{e}^{-i\omega t} C_{ij} (t)\right|^2 \end{eqnarray} $  (2)

$ \begin{eqnarray} C_{ij}(t)& = &\langle \delta E_{ij}(t')\delta E_{ij} (t-t')\rangle_{t'} \end{eqnarray} $  (3)

Here, the brackets represent canonical averaging. FIG. 3 shows that low-frequency vibrations ranging from 30 cm^-1 to 300 cm^-1 dominate the spectra, including the phonon modes coming from the inorganic and organic components. In the case of pristine FAPbI₃, the main peak around 100 cm^-1 is attributed to the Pb-I stretching and bending modes [56]. These modes couple to charge carriers, create the NA coupling, and lead to electron-hole recombination. The peaks between 100-200 cm^-1 and 200-300 cm^-1 are assigned to the libration and torsional of the organic cations [56], which contribute to the NA coupling via the electric field they created. Rotating partial FA cations suppress atomic fluctuations and result in the main peak moving to low frequency, which are responsible for not only decreasing NA coupling but also increasing decoherence time, Table Ⅱ.

FIG. 4 shows the pure-dephasing functions,

$ \begin{eqnarray} D_{ij}(t) = \rm{exp}\left(-\frac{1}{\hbar^2}\int_0^t \rm{d}t' \int_0^{t'} \rm{d}t{"} C_{ij}(t{"})\right) \end{eqnarray} $  (4)

which are obtained by double integration of autocorrelation function, $ C_{ij}(t) $, using the second-order cumulant approximation in optical response theory [34]. Fitting the functions to a Gaussian, exp = -0.5$ \times $($ t/\tau $)$ ^2 $, gives the pure-dephasing times, $ \tau $, in Table Ⅱ. The pure-dephasing times are 5-7 fs, which are remarkably shorter than the nonradiative electron-hole recombination times. Thus, it is necessary to consider the decoherence correction in NAMD simulations [57]. In general, short pure-depahsing time favors long electron-hole recombination time. When the pure-deaphsing time approaches to zero, the quantum dynamics stops, as rationalized by the quantum Zeno effect [58].

The unnormalized autocorrelation functions (un-ACFs) of bandgap fluctuation (inset of FIG. 4) provide proof for the difference in the pure-dephasing times. Generally, larger initial value and faster decay of un-ACFs lead to shorter pure-dephasing time. The un-ACFs of two systems decay on similar periods. As a result, the pure-dephasing time difference is determined by the the initial value of the two un-ACFs. Shown in the inset of FIG. 4, the initial value of the un-ACF for the pristine system is about 1.5 times larger than that in the R-FA system, rationalizing the short pure-dephasing time in the pristine system.

FIG. 5 shows the nonradiative electron-hole recombination dynamics of the pristine and R-FA systems. The recombination time scales, $ \tau $, summarized in Table Ⅱ, are obtained by fitting the population decay functions to an exponent, $ P(t) $ = exp($ -t/\tau $). The electron-hole recombination time scale in the R-FA system is 3.17 ns, which occurs within sub-nanosecond in pristine system and is consistent with the the experiment [24]. As above-mentioned, the small difference in bandgap has almost no influence on the difference in electron-hole recombination time scales for the two systems. Therefore, the recombination time difference is determined by the NA coupling and the pure-depahsing time. According to the Fermi's golden rule, the recombination time rate is proportional to the NA coupling square. The decoherence has a subtle influence on the electron-hole recombination rate. The NA coupling in the R-FA system is twice larger than that in the pristine system, leading to a 3.9 times longer electron-hole recombination time (Table Ⅱ). The pure-dephasing time increases by a factor of 0.2 of the R-FA system with respect to the pristine FAPbI₃, which increases the recombination time scale. Consequently, the smaller NA prevails the longer pure-dephasing time, slowing the electron-hole recombination and extending the excited-state lifetime to 3.17 ns. The long-lived excited charge carries are beneficial for optimizing the performance of perovskite solar cells and optoelectronic devices by reducing charge and energy losses.

By performing TD-DFT combined with NA-MD simulations, we have investigated the influence of the rotation of FA cations on the nonradiative electron-hole recombination in FAPbI₃ perovskite and established the factors for the notably extended excited-state lifetime in the rotated structure. A mechanistic understanding of the nature for the retarded electron-hole recombination is of great importance for design of efficient FAPbI₃ perovskite solar cells. Promoted by I-Pb stretching and binding modes, FA rotation delays nonradiative electron-hole recombination by a factor of 3.9 compared to FAPbI₃, due to the notably reduced NA coupling. The NA coupling is reduced because the rotation of FA cations distorts the inorganic lattice around the rotated region and suppresses the atomic motions, in which the former factor decreases the NA coupling by virtually localizing the electron and hole wave functions on different locations. The latter factor reduces NA coupling due to decreased nuclear velocity but which increases the decoherence time. Consequently, small NA coupling competes successfully with the long decoherence time, extending the charge carrier lifetime to 3.17 ns in the rotated structure, which is about sub-nanosecond in pristine system. The obtained time scales agree well with the experiments. The study advances our understanding of the excited-state properties of perovskite and generates valuable insights for control of charge-phonon dynamics for improving the device performance.

This work was supported by the National Natural Science Foundation of China (No.21573022 and No.51861135101). Run Long acknowledges the Recruitment Program of Global Youth Experts of China and the Beijing Normal University Startup.