A new perspective on lone pair dynamics in halide perovskites

Halide perovskites form the foundation of an emerging class of materials for broad application in renewable and sustainable applications, including photocatalysis and solar energy harvesting. These materials exhibit beneficial photophysical properties, including bandgaps suitable for solar energy harvesting and efficient charge screening that underlies efficient charge carrier separation and resistance to defects. For organic–inorganic hybrid perovskites, these benefits are thought to arise, in part, from dipolar molecular cations that can reorient in response to charged particles and defects. In this work, we provide a similar perspective for inorganic metal halide perovskites, which do not contain molecular species with permanent dipoles. We discuss how lone pair electrons lead to dipolar ions that exhibit dynamics in analogy with traditional molecular plastic crystals and hybrid perovskites. We argue that further understanding these electronic plastic crystal motions with first principles simulations and synchrotron scattering can help create a basic understanding of photophysical properties of metal halide perovskites and inform the design of advanced functional materials. © 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0001908., s


I. BACKGROUND
Metal halide perovskites of generic composition ABX 3 have garnered significant attention as advanced materials for renewable and sustainable energy applications. [1][2][3] For example, numerous halide perovskites show exceptional photovoltaic efficiency, promising improvements in solar cell technologies and use as components in photocatalytic devices. 3 The favorable optical and electronic properties of halide perovskites have created interest in their use in optoelectronics, such as light emitting diodes (LEDs). 2,4,5 Much work has focused on the structure and dynamics of hybrid organicinorganic halide perovskites, composed of an organic cation at the A-site, a metal at the B-site, and halides at the X-sites. For example, methylammonium lead iodide, NH 3 CH 3 PbI 3 or MAPbI 3 , is a highly studied member of this class of perovskites. [1][2][3]6 Unlike simpler A-site cations, such as Cs + , the MA + cation has a permanent dipole moment that can undergo thermally induced rotational motion. [7][8][9][10][11][12] Moreover, upon introduction of an excess charge into the lattice, e.g., an atomic defect site or an excess electron, these dipolar cations can readily reorient to polarize in response and screen the charge. 11,13 Such liquid-like screening is thought to underly the robustness of halide perovskites to defects because charged defects are readily screened by this polarization. [13][14][15][16][17] This screening is also suggested to result in the ability of halide perovskites to efficiently separate charge carriers from bound excitons, ultimately by "solvating" each diffuse charge, much like a polar solvent solvates and separates ions of opposite charge. 11,[13][14][15]17 Both simulations and experiments have shown that these MA + dipoles have a rotational correlation time on picosecond timescales, leading to fast polarization of the perovskite material in response to charged excitations. [7][8][9][10] Characterizing the dipolar structure and dynamics of organicinorganic halide perovskites has created a foundation for understanding the variety of important photophysical properties of these materials. However, similar properties are observed for other halide perovskites with simpler A-site cations, such as CsPbX 3 and CsSnX 3 , where X indicates Cl, Br, or I. [18][19][20][21] For example, inorganic halide perovskites can also have high dielectric constants and efficiently separate and solvate charges, much like the organic-inorganic hybrid perovskites. Moreover, both the simulation and experiment have identified large and fast (sub-ps) polarization fluctuations in this class of metal halide perovskites. 22 These observations suggest that CsPbX 3 , CsSnX 3 , and similar perovskites behave qualitatively analogous to those containing dipolar organic cations, despite the noticeable absence of a permanent dipole-containing molecular ion. This suggests that the permanent dipole of an organic cation is not solely responsible for the advantageous photophysical properties of halide perovskites, but instead, the inorganic framework can dominate the behavior of these materials. [18][19][20][21] At ambient temperatures, CsPbX 3 and CsSnX 3 halide perovskites typically are in the cubic phase, shown in Fig. 1(a) for CsSnBr 3 . However, Seshadri and co-workers have shown that the B-site ion is not located at the center of an octahedral coordination environment in the cubic phase, as one might expect. [23][24][25][26][27] Instead, Pb and Sn ions exhibit dynamic off-centering; the ion position fluctuates between eight energetically favorable asymmetric configurations. Ultimately, this off-centering stems from the s 2 lone pair of each of these cations and is not observed for symmetric B-site cations, as in CsCaX 3 perovskites. Localization of this lone pair (LP) results in a Pb-LP or Sn-LP dipole, and it is energetically unfavorable for the lone pair to point toward one of the surrounding X ions due to electrostatic repulsion, as shown in Fig. 1(b). These interactions push the B-site cation off-center, which manifests in the pair distribution function, g(r), shown in Fig. 1(c), as predicted by simulations described in Sec. II. For CsSnBr 3 , the Sn-Br g(r) first peak is asymmetric due to the asymmetry of the off-center Sn ion. In contrast, halide perovskites with symmetric B-site cations without a significant dipole, such as CsCaBr 3 , display a symmetric first peak in the Ca-Br g(r) [ Fig. 1(c)]. To minimize electrostatic repulsions between like charges, the Pb or Sn atom will move away from the center of its octahedral coordination shell and point its dipole toward one of the eight faces of the octahedron.
At temperatures large enough to cause cation off-centering, e.g., above the phase transition temperature to the cubic phase of CsSnBr 3 , 24 thermal fluctuations activate rotational motions of the lone pairs in the system. 28 This type of electronic plastic crystal motion can be quantified by monitoring time correlations of relevant descriptors. For Sn-LP dipole rotational motion, a good descriptor is the normalized Sn-LP dipole moment vector. For Cs + or Br − , each with four orientationally equivalent lone pairs, on average, the descriptor must account for the orientational symmetry of the local electronic structure. Recent work has used tetrahedral rotor functions to describe the orientational fluctuations of lone pairs around these ions, 28 taking inspiration from the examination of molecular plastic crystals. 29,30 Here, we discuss this perspective further and create an analogy between dipolar cations such as MA + and the Sn-LP dipoles in metal halide perovskites.

II. COMPUTATIONAL METHODS
We perform ab initio molecular dynamics (AIMD) simulations of halide perovskites using CP2K and the QUICKSTEP module. 31,32 We employ molecularly optimized (MOLOPT) Goedecker-Teter-Hutter (GTH) double-ζ valence single polarization short-ranged (DZVP-MOLOPT-SR-GTH) basis sets 32 with GTH-PADE pseudopotentials 33 to represent the core electrons. We explicitly treat the valence electrons using the PBE 34 functional with a plane wave cutoff of 400 Ry, as implemented in CP2K. This choice of functional enables connection to earlier work on similar systems. 24 We first equilibrated each system, consisting of a 3 × 3 × 3 supercell, at constant temperature using a Nosé-Hoover thermostat chain of length three 35,36 and the equations of motion were integrated with a timestep of 1.0 fs. We then equilibrated each system in PERSPECTIVE scitation.org/journal/apm the microcanonical (NVE) ensemble. Production simulations, over which statistics were gathered, were performed in the NVE ensemble over a minimum of 10 ps of simulation time. The maximally localized Wannier function centers (MLWFCs) were computed using CP2K, wherein the MLWF spreads were minimized according to the formulation of Ref. 37. We characterize the rotational dynamics of MLWFCs with two time correlation functions, discussed in previous work. 28 In order to quantify the Sn-lone pair dipole rotational motion, we present the rotational correlation function where μ(t) is the Sn-MLWFC unit vector at time t and P 2 (x) is the second order Legendre polynomial. For Br atoms, we discuss the TCF of tetrahedral rotor functions, Mγ, of order l = 3, which have been used to characterize the rotational motion of tetrahedral structures in previous work on ionic crystals. 29 Here, γ labels the (2l + 1) functions for each l, and we focus on γ = 1 here for simplicity, with other relevant values of γ yielding similar results. 28 In the case of γ = 1, the tetrahedral rotor function is given by where ri = (xi, yi, zi) is a unit vector along a Br-MLWFC bond i. We then quantify the rotational motion of Br MLWFCs through the TCF, where δM 1 (t) = M 1 (t) − ⟨M 1 ⟩. Monitoring unit vector correlations strictly probes rotational motion of the vectors, ignoring changes in Br-MLWFC bond length, arising from changes in covalency. However, using the unnormalized Br-MLWFC vectors leads to nearly identical C 1 (t) correlation functions.

III. RESULTS AND DISCUSSION
The timescale for rotational motion of lone pairs can be quantified through the time correlation functions mentioned above: the Sn-LP dipole rotational correlation function and the time correlation function of the tetrahedral rotor functions for Br-LP correlations. 28 Both are shown in Fig. 2(a) for CsSnBr 3 . These time correlation functions indicate that lone pair rotational motion of all sites occurs on timescales of several hundred femtoseconds. 28 This timescale is the same as that observed and predicted for polarization fluctuations in these materials, suggesting that lone pair rotations may be tied to these polarization fluctuations, especially those of the Sn-LP dipoles. 22 Importantly, because the Br-MLWFC rotations involve changes in Br covalency, the decay of Cγ =1 (t) includes any bond breaking and therefore places an upper bound on this timescale.
The prediction of rapid reorientational motion of dipolar cations suggests that many of the concepts developed for dipolar organic cation-containing perovskites can be extended to those containing a dipolar B-site cation, except that the dipole is more dynamic (its magnitude fluctuates more easily) and its rotational motion is faster. Moreover, perovskites that contain two dipolar species, such as MAPbBr 3 , will create additional local static and dynamic correlations due to the interactions between MA + and Pb 2+ -LP dipoles. 25 We suggest that the Sn-LP dipoles are connected to many of the favorable screening characteristics of CsSnX 3 and similar perovskites and can be examined in a manner similar to permanent MA + dipoles. 14,15,22,38 The electronic plastic crystal motions in halide perovskites have implications for understanding thermal fluctuations in the electronic structure. Previous work has shown that off-centering of Sn ions widens the bandgap of CsSnBr 3 . 24 Thus, one may anticipate that translational and rotational fluctuations of the Sn 2+ -LP dipole, as well as the Br MLWFCs, will lead to fluctuations in the bandgap,

PERSPECTIVE
scitation.org/journal/apm creating a distribution of energies at which CsSnBr 3 can be excited. Additionally, the top of the valence band of CsSnBr 3 is dominated by contributions from Br. [39][40][41] Therefore, we expect that fluctuations in the valence band energy will arise from fluctuations in the electronic structure of Br 42 such that the decay of the MLWFC rotational correlations will occur on timescales similar to those characterizing valence band energy fluctuations. Similar, arguments hold for fluctuations in the conduction band energy because the bottom of the conduction band is also dominated by the B-site metal orbitals. [39][40][41] We can quantify the timescale for valence band fluctuations through the decay of the time correlation function, where δεv(t) = εv(t) − ⟨εv⟩ and εv(t) is the energy eigenvalue of the valence band at time t. We also define analogous time correlation functions for the conduction band energy eigenvalue, εc(t), and the bandgap, Eg. These time correlation functions are shown in Fig. 2(b) and decay on a similar but slightly faster timescale than the electronic rotational motion of the lone pairs. This faster decay may be expected based on the sensitivity of the valence and conduction band energies to changes in the covalency of Br atoms. As mentioned above, the decay of Cγ(t) places an upper limit on Sn-Br bond lifetimes. Therefore, the faster decays of Cv(t), Cc(t), and CE g (t) are consistent with Br exhibiting changes in covalency on a timescale less than the rotational timescales.
To further this perspective we examine time correlations of Sn-Br covalent bonds by monitoring nuclei-nuclei and nuclei-MLWFC correlations, following recent work. 28,[43][44][45] We focus on strongly covalent bonds, consistent with relatively short Sn-Br bond lengths in the perovskite. Thus, we define a covalent bond to exist between a Sn-Br pair when the Sn-Br distance is less than 2.75 Å, the Sn-MLWFC (of Br) distance is less than 2.4 Å, and the Sn-Br-MLWFC angle is less than 20 ○ . The linear Sn-Br-MLWFC angle is consistent with an electron pair located between the Sn and Br atoms, while the Sn-MLWFC distance cutoff excludes Br lone pairs, which are further from the Sn site due to increased localization on the Br atom [see Fig. 1(b) for an illustration]. We then define an indicator function, h(t), which is equal to one when an Sn-Br bond exists at time t and zero otherwise. Finally, the lifetime of strong Sn-Br covalent bonds can be monitored through the time correlation function, where δh(t) = h(t) − ⟨h(t)⟩. The time correlation function C bond (t) is also shown in Fig. 2(b) and decays on the same timescale as the electronic structure fluctuations. This suggests that the valence and conduction band fluctuations are indeed dominated by changes in Br covalency. We note that including increasingly weaker Sn-Br bonds in C bond (t), by extending the Sn-Br cutoff distance, increases the decay time, spanning that shown in Fig. 2(b) to that for Crot(t) and Cγ(t), when the weakest bonds are included.

IV. CONCLUSIONS
In this Perspective, we suggest that localized lone pair electrons in metal halide perovskites exhibit rotational motion analogous to nuclear motions in molecular plastic crystals. We expect that these electronic plastic crystal dynamics can be observed in synchrotron measurements of the fluctuating charge density, [46][47][48][49] as well as spectroscopic methods that can probe dynamics on femtosecond timescales. 50,51 The rotational dynamics are tied to fluctuations of the electronic band energies through changes in Br covalency coupled to MLWFC rotations, with the timescale for fluctuations in valence band, conduction band, and bandgap energies being slightly faster than those for the electronic rotational dynamics. Indeed, the valence and conduction band are dominated by Br and Sn states, respectively, 41 and the dynamics of their electronic structure will impact time-dependent fluctuations of the band energies.
Our perspective on the local electronic structure and dynamics in metal halide perovskites may aid in understanding photophysical and photochemical properties of these materials. For organicinorganic hybrid perovskites, characterization of the structural and dynamic properties of dipolar organic cations has led to significant advances in our understanding of the favorable properties of these materials. [1][2][3][6][7][8][9][10][11][12][13][14][15][16][17] We expect that similar qualitative understanding can be developed by generalizing these insights to dipolar metal cations, such as Sn 2+ , in halide perovskites.
The ability of lone pair electrons to rotate at finite temperature in response to polarization fluctuations may also be relevant for understanding the surface chemistry of materials. At perovskite surfaces, lone pair sites can be exposed. 52,53 Rotational dynamics of these lone pairs could impact the structure of a perovskite interface (with a liquid, for example) and the corresponding interfacial chemistry.
We close by noting that this perspective on lone pair dynamics is not limited to halide perovskites, and we expect electronic plastic crystal motion of lone pairs to be prevalent in solids with localized lone pair electrons at finite temperatures. For example, we have predicted that these charge fluctuations should exist in halide crystals, 28 and future work will generalize these ideas to additional halide-containing solids.

ACKNOWLEDGMENTS
This work was supported as part of the Center for Complex Materials from First Principles, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0012575. Computational resources were supported, in part, by the National Science Foundation through major research instrumentation Grant No. 1625061 and by the U.S. ARL under Contract No. W911NF-16-2-0189. We thank Ram Seshadri, Umesh Waghmare, and Tony Cheetham for useful discussions and generously sharing their thoughts and work on this subject.

DATA AVAILABILITY
The data that support the findings of this study are available within the article. Raw data files are available from the corresponding author upon reasonable request.