Nonadiabatic molecular dynamics analysis of hybrid Dion–Jacobson 2D leads iodide perovskites

Conventional three-dimensional (3D) halide organic–inorganic perovskites (HOIPs) have achieved solar cells with power conversion efficiencies (PCE) exceeding 25%.^1,2 However, they are often unstable to moisture, light, and heat.^3–7 3D HOIPs are also limited in the choice of organic moiety due to structural stability as defined by the empirical Goldschmidt tolerance factor.^8,9 In contrast, two-dimensional (2D) HOIPs do not face the same intrinsic restriction for choosing spacer cations. Consequently, 2D perovskites can incorporate bulky organic cations [butylammonium, $BA+$⁠, phenethylammonium, $PEA+$⁠, and 3-(aminomethyl) piperidinium, 3AMP, 4-(aminomethyl) piperidinium, 4AMP] that act as the spacer molecules and provide a steric barrier for surface water intrusions.^10–12 With versatile tunings of interlayer cations endowing these 2D perovskites with rich optoelectronic properties,^13–16 they have become a promising class of materials for solar cells and other optoelectronic applications. Despite improved structural properties, initially, 2D HOIPs struggled to compete with other high-performing optoelectronic materials, due to reduced and anisotropic charge transport,^17,18 caused by the bulky organic spacer layers. However, an intense research effort has partially addressed this limitation and the PCE of 2D perovskites has improved from 4.73% (Ref. 19) to 18% (Refs. 20 and 21) recently. The enhanced efficiency of 2D halide perovskite devices results from better intrinsic properties of the materials, which are systematically engineered during and after synthesis.^20,22,23 Further improvements in the carrier dynamics and PCE will require an in-depth understanding of these materials that can be supported by ab initio investigations.

Currently, 2D Ruddlesden–Popper^14,24 (RP) and Dion–Jacobson^25–28 (DJ) phase perovskites are two promising classes of materials for optoelectronic devices. RP phases have been around for decades and are easier to synthesize. DJ phase perovskites are much newer and harder to synthesize; however, they have realized an explosive growth of PCE from 7.32% (Ref. 29) to 15.6% (Ref. 30) within only one year, indicating they are a more compelling path forward. Both phases exhibit high moisture resistance and tunable optoelectronic properties.^31,32 2D perovskites are usually defined with the formula (⁠$A′$⁠)_m(A)_n−1B_nX_3n+1, where $A′$ can be divalent (m = 1) or monovalent (m = 2) cation that forms a bilayer or monolayer connecting the perovskite (A)_n−1B_nX_3n+1 2D sheets, where n indicates the layer thickness of metal halide sheets that can be adjusted by tuning precursor composition. RP and DJ phase perovskites have m = 2 and m = 1, respectively. “A” represents a short monocationic molecule or atom (e.g., Cs⁺), “B” is the divalent metal atom (Pb, Sn, Ge), and “X” is the halide (Cl, Br, I). The structural difference between these two phases is shown in a relative stacking of the layers: the RP phase perovskites have two monovalent spacer cations. Here, due to the large interlayer distance and weak van der Waals interaction between spacer monocations, the interlayer conductivity is severely restricted. In contrast, the interlayer gap is bridged by only one short bivalent spacer cation in DJ perovskites, where the inorganic layers stack exactly on top of each other and the van der Waals gap between adjacent interlayer cations disappears. This introduces fewer degrees of freedom into the structure, making the inorganic layers closer to each other, and resulting in better out-of-plane charge transport properties and improved stability.³³ Recent report also exhibits a smaller bandgap for the same value of n and a shorter interlayer distance for DJ-phased perovskites compared to the RP phase. Despite showing more potential for optoelectronic applications compared with the RP phase perovskites, due to both DJ phase's newness to the community and their more difficult synthesis, systematic investigations into the fundamental aspects and systematic investigations into the optimization of DJ phase perovskite devices are nearly untouched.

Here, we perform an in-depth comparison of structural and electronic properties that dominate the photophysical properties of two different DJ phase perovskites: (3AMP)$(MA)3Pb4I13$ and (4AMP)$(MA)3Pb4I13$⁠. To consider dynamical processes influencing charge carriers at ambient conditions, we employ ab initio real-time time-dependent density functional theory (DFT) and nonadiabatic molecular dynamics (NAMD) modeling. We find that different spacer cations significantly impact the thermal structural variations in these halide perovskites. (3AMP)$(MA)3Pb4I13$ has larger structural fluctuations causing faster quantum dephasing compared to (4AMP) $(MA)3Pb4I13$⁠. Comparable nonadiabatic coupling with faster quantum dephasing leads to a long carrier lifetime in 3-AMP-based perovskites compared with 4-AMP-based perovskites. In addition, electron–phonon coupling in DJ phases may have a decisive impact on the reduction of extrinsic nonradiative recombination and carrier trapping as shown previously for RP phases.³⁴ 

We start our simulations with previously reported structures²⁹ as the initial geometries for (3AMP)$(MA)3Pb4I13$ and (4AMP)$(MA)3Pb4I13$⁠. In these geometries, every slab with four layers of $[PbI6]4−$ octahedra remains separated by a layer of 3AMP or 4AMP dications. We further refer to these materials as 3AMP and 4AMP perovskites, respectively. Due to the presence of light atoms and thermally activated molecular dynamics, x-ray diffraction-based structural determination techniques cannot discern the exact conformation of the asymmetric organic cations in between the two inorganic layers with absolute certainty, as well as structural distortions related to inversion symmetry breaking. As a result, recent studies have reported two different interlayer cation alignments in the spacer region.^35,36 Thus, we initially consider both spacer-cation conformations [Figs. 1(c)–1(f) and supplementary material Fig. S1] because these two conformations may coexist. In these two conformations, the nearest-neighbor spacer cations (i.e., 3AMP or 4AMP molecules) along the b-axis remain aligned [Figs. 1(c) and 1(d)] or anti-aligned [Figs. 1(e) and 1(f)] to each other in the ab-plane. Our structure optimizations (see the supplementary material for Computational Methods) with fixed cell parameters and without any constraints on atomic positions reveal that for both perovskites, the aligned and anti-aligned structures are energetically very similar (supplementary material Table S1). The experimental bandgap values²⁹ of these two perovskites are 1.87 and 1.89 eV, respectively. Simulated bandgap values without considering spin–orbit coupling (SOC) of aligned structures and anti-aligned structures are shown in Table I. The bandgap values without SOC correction are 10's of meV away from the experimentally measured data for the 4AMP system. This seemingly excellent agreement is actually due to a well-documented mutual cancelation of errors that stems from leaving out spin–orbit coupling^37–42 (bandgap narrowing) and electronic many-body interactions⁴³ (bandgap widening). Unfortunately, with 260 atoms per simulation cell, the inclusion of SOC is prohibitive for many of the calculations in this work (see the supplementary material for Computational Methods). Nonetheless, with or without the inclusion of SOC correction, we notice two important trends from our simulations. First, the calculated bandgap is largely insensitive to the cation orientation in the case of the 4AMP perovskite; however, the bandgap of 3AMP significantly changes with the alignment of its cations. It is known experimentally that the shape of the 3AMP cation by comparison with its 4AMP counterpart is at the origin of strain accumulation in the corresponding DJ compounds or intermediate n values.^27,28 Second, computational findings for the anti-aligned conformation of 3AMP match closely with the available experimental structural data. To analyze structural features, we rely on previous extensive work relating the backbone structure to the electronic properties of 2D perovskites.^10,15,28,44 First, we note that the average Pb–I–Pb bond angle and Pb–I bond length (supplementary material Table S2) for both 3AMP and 4AMP perovskites do not change significantly between the aligned and anti-aligned structures. In order to quantify how much of the bandgap opening observed for 3AMP (Table I) is cation orientation vs perovskite backbone, we place the anti-aligned cations in the perovskite backbone that was optimized for the aligned 3AMP and optimized only the interlayer organic atoms while keeping the perovskite atoms frozen. The bandgap from this test is 1.84 eV, indicating that the cation orientation is almost entirely responsible for the bandgap opening that matches experiment. Finally, inspection of noncovalent hydrogen-bonding (H-bonding) shows that the average H-bond length between inorganic Pb₄I₁₃ layers and spacer cations is shorter in the anti-aligned conformations (supplementary material Table S3) compared with the aligned ones. This further indicates that the anti-aligned structures have stronger H-bonding interactions that stabilize this conformation.

Since the cation alignment orientation is of interest to the community, and somewhat surprisingly important to obtain consistent basic electronic structure for 3AMP, we explore this factor in more detail. First, we examine the dipole moments for the two molecules (see the supplementary material for Computational Methods). As shown in supplementary material Table S4, when the interlayer cations are aligned, the dipole moments of two nearest-neighbor cations for 3AMP and 4AMP are 9.09 and 6.06 Debye, respectively. In these aligned perovskites, the dominant part of the dipole moments of interlayer cations lies along the out-of-plane direction (c axis), pointing to the aminomethyl. This aminomethyl is more positively charged than $(NH2)+$ within the piperidinium based on the dipole moment direction. When inspecting the partial charge density of the band edge states for the aligned systems, we do observe noticeable asymmetries for 3AMP and 4AMP in supplementary material Fig. S2. 3AMP is very asymmetric within the perovskite layers with the electron and hole wavefunctions being separated. This separation rationalizes the erroneously calculated low bandgap, owing to a known problem of DFT when dealing with charge-separated states.^45,46 When the interlayer cations are anti-aligned, the dipole moment of two nearest-neighbor cations for 3AMP and 4AMP is 1.01 and 4.15 Debye, respectively. For anti-aligned structures, dipoles are directed within the ab-plane, which leads to a more symmetric partial charge distribution (supplementary material Fig. S3). This large difference in the dipole moment between the aligned and anti-aligned two nearest-neighbor 3AMP rationalizes the bandgap mismatch between the aligned conformation and the experimentally measured bandgap in this system. These results demonstrate that to accurately model these asymmetric spacer cation-based systems, it is very important to check their possible mutual orientations, corresponding energetics, and electronic properties. As such for subsequent calculations, we only consider the anti-aligned conformations of both DJ phase perovskites. While we show that for the 3AMP system the cations need to be in an anti-aligned configuration due to limitations of DFT, the actual orientation of the cations in the experiment remains an open question.

Experimentally, 3AMP and 4AMP bandgaps (around 1.9 eV) are in the visible light range. Calculated band structures are shown in supplementary material Fig. S4. I-5p orbitals dominate the valence band maximum (VBM), and Pb-6p orbitals dominate the conduction band minimum (CBM) for both perovskites as illustrated in supplementary material Figs. S3 and S5. There is still a slight orbital asymmetry for the anti-aligned structures in the charge density distributions that is most likely arises from the residual dipole moments of these two systems. This is likely due to variations in the octahedral tilting angles along the stacking axis and preventing perfect inversion symmetry.⁴⁷ To get some insights into the charge carrier transport, we calculated the effective masses from the band edge structures computed including SOC. As shown in supplementary material Table S5, the effective masses of electrons and holes along with the stacking direction range from 0.22 m₀ to 0.63 m₀, which is greater than the effective masses of in-plane carriers (0.17–0.39 m₀). In supplementary material Figs. S2(a) and S2(b), bands near CBM and VBM look almost flat due to the short distance between Γ and B in the reciprocal space. However, if we zoom in to the part of Γ→B (see the supplementary material Fig. S6), we can observe clear curvatures, suggesting that these DJ phase perovskites have better charge transport along the stacking direction compared with other 2D perovskites,⁴⁸ which typically have very large effective masses and almost flat dispersion curves compared with their 3D counterparts along the stacking axis.¹⁵ When we include SOC corrections, two bands near the VBM of 3AMP remain nearly doubly degenerate, Fig. S6(a). The substantially lighter effective hole of one of the bands near the VBM of this perovskite can improve the charge mobility along the stacking direction significantly compared with that of 4AMP.

Once 3AMP and 4AMP are heated up, such that the following ab initio molecular dynamics (AIMD) trajectory remains stable at 300 ± 5 K for 5 ps, we observe several important structural features. From the T = 0 K optimized geometries, we find that the averaged $[PbI6]4−$ octahedra of 4AMP become more distorted compared with that of 3AMP. Further, 3AMP has smaller time-averaged Pb–I–Pb bond angle deviations from 180° over the simulation cell (Table S6). With a temperature increase, there is less octahedral distortion for both materials; however, 4AMP remains the more distorted as seen from the Pb–I–Pb bond angle, supplementary material Table S6. The asymmetry in the VBM and CBM charge densities that were observed at equilibrium balances out at RT, and charge densities become evenly distributed across the first and third sublayers as shown in Fig. 2. Another point to note is the modification of the Jahn–Teller distortions^49,50 in 3AMP. At equilibrium, the average axial Pb–I bond length is shorter than the equatorial Pb–I bond length for both 3AMP and 4AMP materials. At RT, 4AMP remains the same; however, in 3AMP the average equatorial Pb–I bond length becomes shorter than the average axial bond lengths, supplementary material Table S7. This is one indication that while the Pb–I octahedra are more disordered in 4AMP, 3AMP undergoes more obvious structural changes at ambient conditions. In supplementary material Table S2, 3AMP at equilibrium has a shorter average N–H···I bond length than 4AMP, which might be the origin of dynamically enhanced organic–inorganic sublattice coupling. When calculating the root mean square deviation of distances between atoms within the simulation cells, there is much more variation along the MD trajectory for 3AMP, supplementary material Fig. S7. The larger structural dynamical change leads to larger orbital energy fluctuations along the MD trajectory of 3AMP, Fig. 3. This difference in energy fluctuations is relevant to the dephasing time, which we discuss below.

We next perform NAMD simulations to investigate the nonradiative recombination of charge carriers near the band edges of DJ phase perovskites. Here, we only consider the intrinsic nonradiative processes (extrinsic processes, such as doping/defects induced carrier trapping, are not included). Since our simulations are performed around RT, we include all energy states within 0.025 eV from the band edges in our excited-state dynamics algorithm. This includes states ranging from VBM-1 to CBM+3. We first estimate the radiative lifetime (RLT) by applying the Einstein coefficient for spontaneous emission, $A21$ (see the supplementary material Computational Methods for details). At this level of theory, we find that both materials have a radiative lifetime on the order of μs (in agreement with experiments^51,52) and nonradiative lifetimes (NRLTs) on the order 10 ns as summarized in Table II. Thus, we mostly focus on the nonradiative recombination processes in these materials. Note that multiexciton-driven recombinations have not been considered here.

Excited-state dynamics necessarily involve the coupling of electronic and lattice degrees of freedom. Both elastic and inelastic carrier scattering matter during the electron–phonon process at ambient conditions. The elastic electron–phonon scattering assists the quantum decoherence, resulting in the collapse of superposition to a singular state. The significance and importance of decoherence in condensed-phase nonradiative recombination have been emphasized extensively.^53–56 The interaction between the system and environment dampens the nonadiabatic effects and, consequently, controls the transition rates between electronic states.⁵³ The decoherence time between electronic states i and j can be obtained as the pure-dephasing time in optical response theory using the second-order cumulant approximation,⁵³ 

where $Cijt$ is the unnormalized autocorrelation function of thermal bandgap fluctuation $δEij$ between two states i and j along the trajectory. The pure dephasing functions are shown in Fig. 4(a). To calculate the pure dephasing times, we fit these functions with a Gaussian function exp[−0.5 $(tτ)2$]. As shown in Table II, 3AMP has a pure dephasing time of 2.7 fs that is substantially smaller than 4.0 fs calculated for 4AMP, being consistent with the fact that the dephasing times are inversely proportional to the magnitude of the electronic energy gap fluctuations.⁵⁷ Usually, the fast decoherence process slows the electronic relaxation. This phenomenon can be well understood from the quantum Zeno effect, where a quantum transition halts if the decoherence time is infinitesimal.^58–60 Thus, a shorter dephasing time of 3AMP points to a slower intrinsic nonradiative recombination.

Given the importance of electron–phonon scattering processes, the numerical study of the phonon modes that are coupled to the electronic degrees of freedom provides an additional description of excited-state dynamics. The rapid decay of coherence analyzed above suggests that many phonon modes are involved in this process. The spectral density, which is the Fourier transform of the normalized autocorrelation function of the band gaps, identifies the active phonon modes that are coupled to the associated electronic transition. Figure 4(b) shows that for both 3AMP and 4AMP, the interband transition couples strongly to many low-frequency phonon modes that appear under 500 $cm−1$⁠. This is because the band edge states consist of orbitals of heavy atoms, Pb and I, that give a strong contribution. 3AMP generally exhibits a larger magnitude in spectral densities compared with 4AMP, which indicates a stronger interaction between electron subsystem and phonons in 3AMP. The major peak (53 $cm−1$⁠) for the electron–phonon coupling in 3AMP can be assigned to bending of Pb–I–Pb bonds.^49,61 The major peak for 4AMP (13 $cm−1$⁠) can be assigned to the stretching of Pb–I bonds.⁴⁹ 

We now focus on the inelastic electron–phonon scattering that reveals the extent of the exchange of energy between the electronic and vibrational subsystems and consequently affects the recombination processes in materials. The definition of nonadiabatic coupling (NAC)⁶² (⁠$dij·Ṙ$⁠) between i and j states is as follows:

where $φi/j$ are wavefunction of ith/jth state of the system, $Ṙ$ are phonon velocities, and $εj/i$ are energies of ith/jth states. Since the intra-band relaxations typically happen on hundreds of femtosecond timescales, it is reasonable to assume that all excited carriers relax to the band edges before recombining across the bandgap in these two halide perovskites. Generally, the stronger the NAC is, the shorter the nonradiative carrier recombination lifetime is. Thus, we focus specifically on the NAC between the states at the band edge. The root-mean-square (RMS) NAC between the VBM and CBM of 3AMP is 0.21 meV, which is slightly stronger than 0.19 meV calculated for 4AMP. 4AMP has a larger time-averaged bandgap than 3AMP (Table II) and smaller nuclei position fluctuation (Fig. S7). According to Eq. (3), this is a natural explanation for the smaller NAC value of 4AMP. Figure 4(c) shows a plotted histogram with the frequency of occurrence of NAC values of various strengths, where a high NAC value is empirically defined as being greater than 0.1 meV. The area under the curve above 0.1 meV is nearly identical for the two materials in question. The NAC values large enough to induce the transition among states have almost the same occurrence in these two materials, as also shown in supplementary materialFig. 4(c). Thus, NAC does not play a decisive role in the determination of the relative rates of nonradiative processes in these two materials.

Based on the analysis above, we find that 3AMP has a noticeably longer intrinsic nonradiative lifetime when compared with 4AMP, see Fig. 4(d). A faster ground-state population growth means indeed faster nonradiative recombination.^32,63 This difference originates almost exclusively from the difference in the dephasing processes in these halide perovskites. Further, a reduction of an intrinsic nonradiative lifetime may contribute to an overall larger carrier lifetime and V_oc of 3AMP as found in previous experimental work.²⁹ 

In summary, our time-dependent ab initio NAMD modeling work shines some light on how intrinsic electron-lattice dynamics may affect solar cell device performance for 3AMP and 4AMP perovskite devices. In solar cells, the efficiency is strongly relevant to the dynamics of photoexcited charge carriers. It is important to get an estimation of the timescale of electron–hole recombination and pathways using theoretical and simulation aspects. The current NAMD method we adopted in this work provides a good option, which is also applicable to other 2D hybrid organic–inorganic perovskites. The larger structural dynamics of 3AMP lead to more obvious state energy fluctuation near the band edges and further result in a faster loss of coherence. Combined with nearly identical band gaps and nonadiabatic coupling constants, the faster decoherence of 3AMP leads to a longer intrinsic nonradiative lifetime, which might be a partial explanation for the experimentally higher PCE and performance in solar cells of 3AMP. The energy transfer between electronic and vibrational subsystems is substantially weakened and inhibits the energy loss in these two systems, which leads to longer nonradiative lifetimes. Both materials have an extremely long (microseconds) radiative lifetime at the chosen level of theory. The intense structural fluctuation of 2D DJ perovskites accelerates the decoherence process, leading to slower intrinsic nonradiative recombination, which is beneficial for optoelectronic performances. Our simulations further relate the bandgap sensitivity to a presence of a specific organic cation. Yet, other practical aspects are related to the choice of the organic cation including the material growth: 4AMP-based DJ pure phase crystals have been synthesized up to n = 7 like BA-based RP phases.⁴⁷ The easy growth of 4AMP DJ phases with large n by comparison with 3AMP is attributed to a reduced accumulation of strain within the 4AMP-DJ lattices at room temperature for intermediate n values,^27,28 related to a very small lattice mismatch between the n = 1 compound and the 3D perovskite. On the other hand, longer carrier lifetimes may be a natural explanation for the superior photovoltaic performances of 3AMP-based solar cells, which exhibit a larger open-circuit voltage (V_oc) than their 4AMP-based counterparts. In addition, the larger short-circuit current (J_sc) for 3AMP was attributed to an enhanced carrier transport due to shorter interlayer I–I distances and smaller effective masses.²⁹ This leads to insight into ways to engineer even better-performing devices. Possible interlayer cation engineering or design could be finding interlayer cations, which lead to shorter X···X (X = halide) distance³⁰ or shortening the I–I distance by replacing the single-band ring of 3AMP/4AMP with a benzene ring.⁶⁴ Thus, observed atomic-scale insights on the relationship between the dynamic structure and charge carrier recombination processes may help the community to screen a variety of mono- and di-cationic spacer molecules to optimize the optoelectronic properties of 2D-halide perovskites, in addition to constraints related to material growth and heterostructure optimization for device applications. Overall this work provides valuable guiding principles for experimentalists when choosing 2D perovskite materials for optoelectronic applications.

See the supplementary material for detailed computational method, structure, partial (band decomposed) charge densities, density of states, band structures of aligned and anti-aligned 3AMP and 4AMP, data for total energies, Pb–I–Pb angles, N–H···I bond lengths, effective masses for different systems.

The work at Los Alamos National Laboratory (LANL) was supported by the LANL LDRD program (S.G.L., F.L., W.N., A.J.N., and S.T.). This work was done in part at Center for Nonlinear Studies (CNLS) and the Center for Integrated Nanotechnologies (CINT), a U.S. Department of Energy and Office of Basic Energy Sciences user facility, at LANL. This research used resources provided by the LANL Institutional Computing Program. LANL is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. 89233218NCA000001. J.E. acknowledges the financial support from the Institut Universitaire de France. Y.W. and O.V.P. acknowledge support of the U. S. Department of Energy, Grant No. DE-SC0014429.