J.
Even
*^{a},
M.
Carignano
^{b} and
C.
Katan
^{c}
^{a}Fonctions Optiques pour les Technologies de l'Information, FOTON UMR 6082, CNRS, INSA de Rennes, 35708 Rennes, France. E-mail: Jacky.Even@insa-rennes.fr
^{b}Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, Doha, Qatar
^{c}Institut des Sciences Chimiques de Rennes, ISCR UMR 6226, CNRS, Université de Rennes 1, 35042 Rennes, France
First published on 26th November 2015
The complexity of hybrid organic perovskites calls for an innovative theoretical view that combines usually disconnected concepts in order to achieve a comprehensive picture: (i) the intended applications of this class of materials are currently in the realm of conventional semiconductors, which reveal the key desired properties for the design of efficient devices. (ii) The reorientational dynamics of the organic component resembles that observed in plastic crystals, therefore requiring a stochastic treatment that can be done in terms of pseudospins and rotator functions. (iii) The overall structural similarity with all inorganic perovskites suggests the use of the high temperature pseudo cubic phase as the reference platform on which further refinements can be built. In this paper we combine the existing knowledge on these three fields to define a general scenario based on which we can continue the quest towards a fundamental understanding of hybrid organic perovskites. With the introduction of group theory as the main tool to rationalize the different ideas and with the help of molecular dynamics simulations, several experimentally observed properties are naturally explained with possible suggestions for future work.
Fig. 1 Room temperature properties of hybrid perovskites viewed as a merging of properties of all-inorganic perovskites, conventional semiconductors and plastic crystals. |
Such an orientational degree of freedom is also reminiscent of plastic crystals.^{14,19,20} Indeed, liquid and plastic crystal phases are possible intermediate stages between ordered crystal and liquid phases. Liquid phases exhibit both translational and orientational/conformational disorder, whereas in a plastic crystal phase atoms and the molecular centers of mass (CM) are located at well-defined positions within the periodic lattice. If the disorder is mainly related to molecular rotations rather than molecular conformations, the phase is frequently referred to as a “rotor phase”. In that sense, there is clear experimental evidence of the rotational motion of the organic cations in the high temperature pseudo-cubic phase of HOP, namely from NMR, dielectric measurements, millimeter wave spectroscopy and neutron diffraction.^{21–24} Although a couple of scenarios for the orientation of the molecular cation are considered, based on the early work by Onoda et al.,^{22} there is yet no theoretical framework that provides an in-depth uniform analyses.
In this paper, after a brief introduction to AIP and HOP, we will first revisit the available experimental data related to phonons in AIP and put them into perspective with HOP for which similar data is still lacking. Next, we will subsequently develop two approaches that allow taking the stochastic molecular degrees of freedom into account. The first simplified approach is based on the concept of pseudospins (PS), recently introduced by one of us,^{19} which offers a discrete representation of possible random orientations of the molecular cation accounting for both its elastic and electric multipoles. We then examined how rotator functions, which provide continuous variables, can contribute to rationalize the properties of HOP. To complement the available experimental data, we performed molecular dynamics (MD) simulations at several temperatures and for several super-cells. This work presents for the first time MD simulations^{25–28} combined with group theory analysis^{19} applied to HOP. All these different approaches, when implemented with the support of the group theory, are shown to help streamline the couplings between stochastic molecular excitations and lattice vibrations, the structural phase transitions and possible orientational glassy behavior of perovskite structures.
The preparation of the systems was made using the available crystallographic data, obtained at 400 K for the cubic phase of MAPbI_{3}. Different systems were prepared by the replication of the unit cell for a larger quasi-cubic simulation box. System-222 (96 atoms) contains 2 unit cells in each Cartesian direction. Analogously, larger systems having up to 5 unit cells per direction were prepared. The largest case, System-555, contains 1500 atoms. All the systems were tempered by a succession of NVE simulations to introduce a representative thermal disorder before starting the NVT simulations, which were integrated for at least 40 ps in each case.
Fig. 2 (a) Simple cubic structure of CH_{3}NH_{3}PbBr_{3} at high temperature.^{45} The center of mass (CM) of the CH_{3}NH_{3} (MA) cation is represented by the large green sphere, and lead atoms by grey spheres. Bromide atoms exhibit anisotropically disordered motions as indicated by the brown ellipsoids. Site labels are also indicated. (b) NaCl-type structure of NaCN at high temperature. The sodium cations are represented by large yellow spheres, the CM of cyanides by small light green spheres. (c) Reciprocal space 3D view of the brillouin zone (BZ) with conventional labels for points of high symmetry (Γ, R, M, X). |
Here we will mainly discuss MA-based 3D HOP, but the methodology can be applied for other organic cations having a different geometry, symmetry or polarity. Noteworthily, the replacement of the molecular CM by a simple cation (Na^{+} or Cs^{+}) mimics the influence of ionic interactions on the electronic band diagrams close to the band gap.^{12–14} However, it is necessary to go beyond this simple model in order to describe the crystal packing and structural instabilities, as well as the complex coupling between the disordered orientational dynamics and lattice vibrations. The first step of our approach consists of decoupling translational and rotational degrees of freedom of the cation A. These degrees of freedom can be analyzed separately yielding, for the translational part, a common vibrational analysis for both AIP and HOP bulk solids. The rotational part must be treated carefully to account for the possible disorder effect and will be investigated in a second step. Then rotation–translation coupling allowed by symmetry in the cubic phase will be considered in a third step. Our analysis strongly relies on the use of group theory^{39,40} (see for example the pedestrian guide in ref. 19) and reciprocal space.^{11,43} The labels of the main high symmetry points related to the first Brillouin Zone (BZ) of the Pmm space group are shown in Fig. 2c. For the irreducible representation (IR), we will mainly use Miller and Love notations.^{19,40}
In order to analyze the symmetry properties of lattice vibrations (phonons) belonging to a Pmm structure, 5 atoms (among which 1 is a pseudo atom for the molecular CM in HOP) must be considered.^{11,19,42} The three acoustic modes belong to the Γ_{4}^{−} IR whereas the twelve optical phonons correspond to the 3Γ_{4}^{−} + Γ_{5}^{−} IR (Table 1). At the R and M points of the BZ, the optical mode IR decompositions yield R_{1}^{+} + R_{3}^{+} + R_{4}^{+} + 2R_{5}^{+} + R_{4}^{−} and M_{1}^{+} + M_{2}^{+} + M_{3}^{+} + M_{4}^{+} + M_{5}^{+} + M_{2}^{−} + 2M_{3}^{−} + 3M_{5}^{−}, respectively. Noteworthily, neither the metal nor the A atom/CM is involved in the M_{3}^{+} and R_{4}^{+} degenerate phonon modes, but only the X atoms. These modes related to antiferro-distortive instabilities correspond to the rotations of the metal halide octahedra. The IR decomposition at Γ are useful to analyze electron to optical phonon coupling. For electron to acoustic phonon coupling, the symmetry properties of the strain tensor components are relevant. The volumetric strain (q_{a} = ε_{1} + ε_{2} + ε_{3}), tetragonal strains (q_{oz} = ε_{1} − ε_{2} and q_{tz} = 2ε_{3} − ε_{1} − ε_{2}) and shear strains (ε_{4}, ε_{5}, ε_{6}) are described by the Γ_{1}^{+}, Γ_{3}^{+} and Γ_{5}^{+} IRs, respectively (Table 1).
Γ | R | M | |
---|---|---|---|
Volumetric strain | Γ _{1} ^{+} | — | — |
Tetragonal strain | Γ _{3} ^{+} | — | — |
Shear strain | Γ _{5} ^{+} | — | — |
Phonons (1a) | Γ _{4} ^{−} | R _{4} ^{−} | M _{3} ^{−} + M_{5}^{−} |
Phonons (1b) | Γ _{4} ^{−} | R _{5} ^{+} | M _{2} ^{−} + M_{5}^{−} |
Phonons (3d) | 2Γ_{4}^{−} + Γ_{5}^{−} | R _{1} ^{+} + R_{3}^{+} + R_{4}^{+} + R_{5}^{+} | M _{1} ^{+} + M_{2}^{+} + M_{3}^{+} + M_{4}^{+} + M_{5}^{+} + M_{3}^{−} + M_{5}^{−} |
PS (A, 1b) | Γ _{1} ^{+} + Γ_{3}^{+} | R _{5} ^{+} | M _{3} ^{+} + 2M_{4}^{+} |
Γ _{4} ^{−} | R _{2} ^{−} + R_{3}^{−} | M _{2} ^{−} + M_{5}^{−} | |
PS (B, 1b) | Γ _{1} ^{+} + Γ_{3}^{+} + Γ_{5}^{+} | R _{4} ^{+} + R_{5}^{+} | M _{1} ^{+} + M_{3}^{+} + 2M_{4}^{+} + M_{5}^{+} |
Γ _{4} ^{−} + Γ_{5}^{−} | R _{2} ^{−} + R_{3}^{−} + R_{4}^{−} | M _{2} ^{−} + M_{3}^{−} + 2M_{5}^{−} | |
PS (C, 1b) | Γ _{1} ^{+} + Γ_{5}^{+} | R _{1} ^{+} + R_{5}^{+} | M _{1} ^{+} + M_{4}^{+} + M_{5}^{+} |
Γ _{2} ^{−} + Γ_{4}^{−} | R _{2} ^{−} + R_{4}^{−} | M _{2} ^{−} + M_{3}^{−} + M_{5}^{−} | |
PS (D, 1b) | Γ _{1} ^{+} + Γ_{2}^{+} + 2Γ_{3}^{+} + Γ_{4}^{+} + Γ_{5}^{+} | 2R_{4}^{+} + 2R_{5}^{+} | M _{1} ^{+} + M_{2}^{+} + 3M_{3}^{+} + 3M_{4}^{+} + 2M_{5}^{+} |
2Γ _{4} ^{−} + 2Γ_{5}^{−} | R _{1} ^{−} + R_{2}^{−} + 2R_{3}^{−} + R_{4}^{−} + R_{5}^{−} | 2M_{1}^{−} + 2M_{2}^{−} + 4M_{5}^{−} |
Unfortunately, similar detailed phonon spectroscopic data and vibrational models are still lacking for 3D HOP. From the available experimental reports on AIP and HOP, we may only infer that the spectra of optical modes lie in the same energy ranges.^{59} Diffraction studies on HOP crystals also report on strongly anisotropic thermal ellipsoids for halide atoms in HOP (Fig. 2a), which can be assigned to a strong anharmonicity.^{45,46} The MA cation reorientations^{21,23,24} have been investigated recently by inelastic incoherent neutron scattering in MAPbI_{3},^{60,61} exhibiting two quasi-elastic responses based on the same technique as used for phonons in AIP. The last neutron scattering investigation^{61} (which appeared while this manuscript was at the revision stage) contains a symmetry analysis of the tetragonal phase related to the approach developed in the present work (vide infra). The MA tumbling dynamics is described by discrete jumps between the four equivalent positions in the tetragonal phase, and a fast dynamics around the C–N axis, which is still present in the low temperature orthorhombic phase.^{61} A recent neutron scattering study of MAPbBr_{3} shows that the low-temperature harmonic (phonon) fluctuations cross over to fast overdamped relaxational dynamics and coupling to molecular dynamics at higher temperatures.^{62} The random reorientations at high temperature thus deserve a specific theoretical framework, which is discussed in the next section.
When plastic crystals are cooled, the rotational motion freezes and an orientational order appears at a certain temperature. Plastic crystals may exhibit two different behaviors at low temperature. In the first case, the rotational motion freezes under careful cooling, keeping quasistatic random molecular orientations. These orientational glasses exhibit most of the properties related to conventional glasses such as specific heat anomalies, ageing and distributions of relaxation times. Prototypes of orientational glasses are cyclohexanol^{64} and cyanoadamantane.^{66} In the second case, the plastic crystal may undergo a structural transition at low temperature to a partially or completely ordered crystalline phase, if spatial correlations or interactions between the molecular orientations are strong enough. Noteworthily, these interactions can be direct or indirect (vide infra). Prototypes of plastic crystals exhibiting structural phase transitions are the NaCN and KCN compounds.^{20} The structural transitions are usually associated to a structural symmetry breaking related to a collective molecular freezing and to an order–disorder character. Displacive and order–disorder dynamical processes are the two limiting cases for the microscopic mechanism driving a continuous structural phase transition. The order parameter dynamics is critical at the transition temperature and for a specific wavevector. For a soft phonon mode and a second order phase transition, the frequency goes to zero at T_{c},^{67} whereas for a critical order–disorder relaxation, a similar behavior is observed for the relaxation rate.^{67,68} The high temperature relaxation mechanism in a plastic crystal is not necessarily critical at the structural phase transition corresponding to the molecular freezing. The order parameter of the structural phase transition can be related to a soft phonon mode of the surrounding lattice. The variable (PS or rotator function) describing the stochastic orientational motion is a secondary order-parameter, linearly coupled to the main displacive order parameter, which leads to a long range ordering of the frozen orientations at low temperature. Long-range correlations may be destroyed in alloys such as (NaCN)_{1−x}(KCN)_{x}, inducing an orientational glass transition at low temperature instead of a structural phase transition.^{69}
HOP compounds undergo structural transitions to low temperature phases, where the molecular orientations are fixed. The existence of stochastic orientational degrees of freedom at high temperature can be assessed from the experimental results reported in the literature. Onoda and coworkers first demonstrated that the specific heat anomalies and entropy contributions at the structural phase transition temperatures are characteristic of order–disorder processes.^{22} Kawamura and coworkers showed that the cubic to tetragonal transition, in MAPbI_{3}, is associated to a partial and collective molecular freezing, together with a displacive-like antiferrodistortive rotation of the metal halide octahedra.^{70} Moreover, Debye-type functions, which are characteristic of relaxational motions, were used to interpret nicely, both dielectric and millimeter-wave spectroscopy results in the three MAPbX_{3} compounds.^{21,23} In the prototype MAPbI_{3} HOP, the quasi-elastic neutron scattering signatures^{60,61} are also consistent with the simultaneous fast rotations of the methyl and ammonium groups around the C–N axes^{24} and the slower tumbling of the C–N axis between the symmetry equivalent positions.^{21,23} This is described in detail in ref. 61 with a C_{4} × C_{3} symmetry-based jump model designed for the tetragonal phase. The same model is used for the cubic phase, although as quoted by the authors, a O × C_{3} model expected to be more correct, yields essentially the same result.^{61} A reasonable description of the experimental results is also afforded by an isotropic orientational distribution.^{61} Both molecular motions are related to entropy contributions at the transition temperatures,^{22} but this work focuses on the tumbling of the molecular axis, namely the C–N axis in MA-based HOP, to discuss PS or rotator functions. The existence of a stochastic reorientational motion, leading to a collective freezing at low temperature, and thus a plastic crystal behavior, seems to be consistent with all available experimental data on MAPbX_{3} compounds. However the nature of the critical mechanism (displacive or order–disorder) is still not clear from the inspection of the literature. This point will be discussed further in section 3.4.4.
The concept of PS is also useful to rationalize properties of other crystals. Table 2 gathers a few examples of plastic crystals showing orientational disorder that can be described by PS in a high temperature cubic phase.^{20} NH_{4}Br and MAPbX_{3} are two examples for a Pmm lattice, whereas NaCN corresponds to a face-centered Fmm cubic lattice. NaCN crystallizes in a NaCl-type structure (Fig. 2b), where the CN^{−} disordered anion occupies the (1/2,1/2,1/2) (A, 1b) site in the cubic cell. The first, second and third nearest neighbors of CN^{−} are Na^{+}, CN^{−} and Na^{+}, respectively. For this reason preferred CN^{−} orientations correspond to {1,0,0} and, to a less extent, {1,1,1} directions. In addition a CN^{−} head to tail disorder is usually assumed, approximating the molecule to a symmetric dumbell.^{20} Finally, in the NH_{4}Br crystal, the tetrahedral molecule in the center has eight nearest neighbor Br anions in the corners. The protons point toward the corners of the cube, which define only two preferred orientations. The NH_{4}Br example further shows that besides the site symmetry, the molecular symmetry is also important.
Substance | Cubic space group |
---|---|
NaCN | Fmm (FCC) |
CH_{3}NH_{3}PbI_{3} | Pmm (SC) |
NH_{4}Br | Pmm (SC) |
In the PS simplified picture, a discrete variable ξ_{i} is defined for each possible molecular orientation i. ξ_{i} is equal to 1 or 0, whether orientation i is occupied or not. Symmetry-adapted combinations of ξ_{i} can be computed for a single crystal cell: . Let's consider scenario A in detail with i = 1–6 for, (1,0,0), (0,1,0), (0,0,1), (−1,0,0), (0,−1,0) and (0,0,−1) orientations respectively. The symmetry-adapted combinations decompose as Γ_{1}^{+} + Γ_{3}^{+} + Γ_{4}^{−} (Table 1):
Γ_{1}^{+}: Ξ_{Γ1+} = ξ_{1} + ξ_{2} + ξ_{3} + ξ_{4} + ξ_{5} + ξ_{6} |
Γ_{3}^{+}: Ξ_{Γ3+} = ξ_{1} − ξ_{2} + ξ_{4} − ξ_{5} and Ξ_{Γ3+}^{2} = −ξ_{1} − ξ_{2} + 2ξ_{3} − ξ_{4}− ξ_{5} + 2ξ_{6} |
Γ_{4}^{−}: Ξ_{Γ4−} = ξ_{1} − ξ_{4}, Ξ_{Γ4−}^{2} = ξ_{2} − ξ_{5} and Ξ_{Γ4−}^{3} = ξ_{3} − ξ_{6} |
The totally symmetric combination Ξ_{Γ1+} is always equal to 1, whatever the value taken by ξ_{i}. In order to better understand the significance of these symmetrized combinations, we may consider the average values of the ξ_{i} in relation to the orientation occupation numbers.^{19,71} If one considers a thermally activated disorder of the molecular orientations, the dynamically averaged value, compatible with lattice symmetry, is <ξ_{i}> = 1/6. It yields 〈Ξ_{Γ1+}〉 = 1, and 0 for the other symmetry-adapted combinations. This can be considered as the ground state of the system in scenario A. The Γ_{3}^{+} and Γ_{4}^{−} IR are relevant to describe dynamically excited configurations corresponding to thermally activated population fluctuations <ξ_{i}> = 1/6 + δξ_{i} with . For instance, the symmetrized combination corresponds to δξ_{1} = −δξ_{4} and δξ_{2} = δξ_{3} = δξ_{5} = δξ_{6} = 0, that is, to a net electric dipole along the x axis. In turn, all the other symmetrized excited combinations yield 0.
The PS concept is able to account for both dynamic and static orientational disorders. Indeed, a molecule with a frozen (static) random orientation and pointing toward a face (ξ_{1} = 1 for example) corresponds to a superposition of Ξ_{Γ1+}, , Ξ_{Γ3+}^{2} and . Static configurations automatically exhibit strong deviations from the dynamically averaged values <ξ_{i}>, which are the only ones fully compatible with lattice symmetries.
Nonpolar even configurations, like Γ_{1}^{+} + Γ_{3}^{+}, correspond to elastic multipoles whereas, polar and odd configurations like Γ_{4}^{−} correspond to electric multipoles (Table 1). A qualitative understanding about the various relaxation modes for the molecular tumbling can be obtained by adapting the relaxation model by Poglitsch et al.^{21} originally designed for scenario C {1,1,1} having 8 possible orientations of the dipole. This model is based on a single relaxation rate γ between adjacent neighboring orientations in a given cubic cell. Noteworthily, it neglects the correlations between neighboring cells. In an extended microscopic description,^{20} the potential energy of the crystal can be developed in the harmonic approximation as a function of translation and rotation variables (PS or rotator functions):
V = V^{R} + V^{TT} + V^{TR} + V^{RR}+⋯ |
For scenario A, each dipole has 4 neighbors. Adapting Poglitsch's model, the system of 6 rate equations for the occupation numbers reads: , where i = 1–6 stands for the 6 possible orientations, and j = 1–4 for the 4 nearest neighbors. Then, the eigenvalues of the relaxation matrix are:
- γ_{0} = 0 for the trivial solution δξ_{1} = δξ_{2} = δξ_{3} = δξ_{4} = δξ_{5} = δξ_{6} (= 0) corresponding to the Γ_{1}^{+} IR
- γ_{1} = 4γ for the triply degenerate solution corresponding to the polar relaxation (Γ_{4}^{−} IR)
- γ_{2} = 6γ for the twice degenerate solution corresponding to the non-polar relaxation (Γ_{3}^{+} IR).
The relaxation rate γ_{1} can be measured by experimental techniques related to electric dipoles such as dielectric measurements,^{21,23} whereas γ_{2} can be probed by experimental techniques relevant to elastic multipoles, like Raman scattering.^{72} Both types of relaxation are expected to yield an experimental signature by incoherent neutron scattering (vide infra),^{60,61} the best technique being triple axis neutron scattering, but large monocrystals are necessary.^{55,56}
Noteworthily, more realistic modelling of the interactions between neighboring cells (vide infra section 3.4.2) may change the γ_{2}/γ_{1} ratio equal to 1.5 in this simplified model. Correlations between the tumbling of cations in neighboring cells also lead to dispersions of relaxation rates. In that case, it is useful to define collective PS variables accounting for the space group symmetry: where k is a wave vector in the BZ, n is for a given molecule at the position X_{n} and N is the total number of molecules. The symmetrized decompositions for these collective variables are given in Table 1 as Γ, R and M for scenarios A–D. Finally, we may consider that the orientational disorder in the Pmm cubic phases of MAPbX_{3} is a superposition of the three scenarios A, B and C. Then, the total number of cells is , where N_{A}, N_{B}, and N_{C} are the number of cells undergoing, respectively, A, B, and C type disorder, respectively. But, as it turns out, such a mixing is better described by rotator functions that will be considered in the next section.
In the cubic phase of MAPbI_{3} at T = 370 K, the experimental analysis of the tumbling motion is difficult.^{61} A fully isotropic random rotation model compatible with the lattice symmetry was used. This continuous model further assumes no correlation between the rotations of the neighbouring molecules. It yields a relaxation rate equal to for the rotation with an angular momentum l (τ = 1.62 ps at T = 370 K). According to this model γ_{2}/γ_{1} is thus equal to 3. By comparison γ_{2}/γ_{1} is equal to 1.5 and 2, respectively, in the Poglitsch discrete jump models for scenarios A (Γ_{3}^{+}/Γ_{4}^{−}, see the above derivation) and C (Γ_{5}^{+}/Γ_{4}^{−}, Table 1). For scenario B, two IR (Γ_{3}^{+} and Γ_{5}^{+} see Table 1) correspond to an angular momentum l = 2, while only Γ_{4}^{−} is relevant for l = 1. γ_{2}/γ_{1} is equal to 2 and 3 in a Poglitsch discrete jump model, respectively, for Γ_{3}^{+}/Γ_{4}^{−} and Γ_{5}^{+}/Γ_{4}^{−}.
Considering a O_{h} molecular site-symmetry,^{19} the probability distribution of the C–N axis orientation as a function of the two polar angles (ϑ,φ), i.e. the unit vector u = (cos(φ)sin(ϑ), sin(φ)sin(ϑ), cos(ϑ)) is given by: where the 〈K_{2i}〉 coefficients are the ensemble average of the cubic harmonics. A cubic harmonic K_{2i} is a specific combination of spherical harmonics of order 2i.^{73} The expansion is usually limited to a few terms and may be used to refine diffraction data. Particular combinations of 〈K_{2i}〉 coefficients are useful to describe a preferred scenario. Noteworthily, pure [001], [110] and [111] orderings are described by infinite series of coefficients and correspond to the limit of PS discrete probability distributions.
By limiting the expansion to the K_{4} term (Table 3, Fig. 3), pure [001] (Fig. 3a) or [111] (Fig. 3b) ordering are qualitatively well described but not the pure [110] one (Fig. 3c), given that the dotted line in Fig. 3c presents a maximum for the [111] direction instead of [110]. It is thus necessary to extend the expansion for pure [110] at least up to K_{6} (Table 3, Fig. 3b). Inclusion of terms up to K_{12} affords a description very close to delta like angular functions for the pure [001], [111] and [110] orientations (see 3D schematic representations shown as inset in Fig. 3). In real systems, fewer terms are usually needed for cubic harmonics expansions. Fig. 4 reports on the experimental results analyzed in the past for NaCN and KCN with an expansion up to K_{6} (Table 3).^{74} For both these materials the [1,1,0] orientation corresponds to an absolute minimum of the probability distribution. NaCN has a preferred orientation along the [001] direction, but significantly different from pure [001] ordering. Indeed, the dashed line in Fig. 4 has a smaller deviation from the horizontal line (corresponding to the isotropic case) than the dashed line in Fig. 3b and the distribution of CN^{−} orientations presents a clear secondary maximum along the [111] direction. The MD simulations performed at 400 and 450 K for MAPbI_{3} are analyzed using the same formalism (Table 3, Fig. 4). The molecular axis exhibits a preferred orientation along [001] but the deviation from the isotropic distribution is less important than in cyanide compounds. This may explain why a reasonable fit of neutron scattering experimental data was obtained in the cubic phase with a purely isotropic rotational model.^{61} Moreover, the [111] direction corresponds to a minimum distribution. This analysis shows that the molecular orientations in MAPbI_{3} correspond to a mixing of A, B and C scenarios, with a preference for scenario A and to a lesser extent for B. In cyanide solids, the [001] and [111] directions correspond to attractive interactions between the CN^{−} and Na^{+} ions (Fig. 2b), whereas a repulsive interaction between MA^{+} and Pb^{2+} is expected along [111] in MAPbX_{3} crystals (Fig. 2a).
Fig. 4 Angular dependence (φ = 45° and θ = [0:90°]) of the probability distribution functions relevant for Γ_{1}^{+} for molecular orientation in NaCN (dashed line) and KCN (dashed-dotted line) limiting the cubic harmonic expansion to K_{4} and K_{6} contributions,^{74} and for MAPbI_{3} (solid line) obtained from MD simulation at 450 K (best fit for K_{6} = 0, Table 3). 3D representations of the corresponding rotator functions are also inserted. |
By analogy to symmetry-adapted combinations that are able to describe PS excited states, it is also possible to design symmetry-adapted rotator functions for rotational excited states. These expansions from spherical harmonics are available for all the IR of the O_{h} group.^{73} They are connected to the symmetry-adapted polynomial expansions provided that the unit vector is expressed as u = (x, y, z). The Γ_{4}^{−} vectorial (l = 1) IR appears in the expansion of the PS for A, B and C scenarios. The lowest-order symmetry-adapted polynomials with Γ_{4}^{−} symmetry appearing as a possible term for a Γ_{4}^{−} rotator function are thus (x, y, z). They correspond to the orientation of the CN axis of MA^{+}. The average time dependence of the autocorrelation function of this vector extracted from MD simulation at 400 K is plotted in Fig. 5 for various cubic supercell sizes.
Fig. 5 (a) Time dependence of the average autocorrelation function of the CN axis orientation in MAPbI_{3} extracted from MD simulation at 400 K for various supercell sizes having an odd or an even number of cubic cells along each direction. (b) Linear chain of p-atomic orbitals of the metal (M(p)) for even and odd numbers of unit cells in the supercell, illustrating, respectively, proper (red oval) and improper (yellow oval) account of the wave function phase factor at the BZ center (Γ) and BZ boundary (R).^{43} (c) Linear chain of MA molecules highlighting the presence of unphysical ferroelectric dipole–dipole interaction (yellow oval) when the supercell is made of an odd number of unit cells. |
Interestingly, the relaxation time depends much on the parity of the number of cells along one direction of the supercell. The CN axis orientation is more stabilized for an even number of cubic cells. The reasons are at least twofold. First, given that the MD simulations are limited to a Γ point sampling of the BZ, hybridization of atomic orbitals related to BZ-folding cannot be adequately addressed with an odd number of cells in one direction. This major effect is exemplified by considering a linear chain in Fig. 5b.^{43} Next, the stabilization of the CN axis obtained with an even number most probably results from enhanced anti-ferroelectric interactions between the molecular dipoles, whereas frustrated interactions are expected with an odd number of cells (Fig. 5c). In the latter case parallel dipole orientations between neighboring supercells may result in unphysical ferroelectric dipole–dipole interactions that are not present in the real bulk crystal. In fact, even though some hybrid perovskites may exhibit phase transitions with a ferroelectric character (vide infra), MAPbI_{3} essentially undergoes antiferro-distortive collective phase transitions at low temperatures, where long-range antiparallel arrangements of the molecular dipoles have been evidenced by diffraction experiments. This highlights the need for appropriate design of supercells in MD simulations, which should not break by construction of the overall symmetry, nor such long-range correlations.
Fig. 6 reports on the temperature dependence of the average autocorrelation function for Γ_{4}^{−} deduced from simulations at three different temperatures. In each case, the relaxation time corresponds to the relaxation rate γ_{1} derived within the PS picture for the tumbling of dipoles. As expected, lowering the temperature induces a slowing down of the molecular tumbling. But, the time dependence is strongly non-exponential and can be nicely fitted using stretched exponentials (Fig. 6): C(t) = e^{−(t/τ)β}. Due to issues related to affordable time scales, it can hardly be extracted at 300 K. For the other temperatures, 400 and 450 K, the stretching parameter β can be reasonably estimated to 0.56. The relaxation time deduced for the electric dipole (l = 1) dynamics (Γ_{4}^{−}), from the simulation at T = 400 K (Fig. 6a), amounts to 10.5 ps. At high temperature, the available experimental data are not consistent: the dynamics is either described by a single relaxation time with a large value of 12.7 ps at T = 370 K^{60} or, a superposition of relaxation dynamics, whose vectorial (l = 1) component has a relaxation time equal to 1.62 ps at the same temperature.^{61}
The Γ_{3}^{+} quadrupolar (l = 2) IR also appears in the expansion of the PS scenarios A and B (Table 1). The lowest-order symmetry-adapted polynomial with Γ_{3}^{+} symmetry appearing as a possible term for an elastic quadrupole is Y^{0}_{2} ∝ (3z^{2} − 1). The relaxation time for the autocorrelation function of this rotator extracted from MD simulation at 400 K is equal to 6.6 ps. It corresponds to the γ_{2} coefficient of the simple PS analysis reported in section 3.3.2. The actual ratio γ_{2}/γ_{1} is equal to 1.6, which is closer to PS scenario A than to scenarios B and C (section 3.3.2). This is also consistent with the maximum observed along the [001] direction for the probability distribution function (Fig. 4). The computed relaxation time is smaller than the experimental one (0.54 ps) deduced from a fully isotropic model for the angular momentum l = 2.^{61}
The analyses of neutron scattering data^{60,61} do not consider lateral coupling between the rotational motions of molecules. Individual molecular random reorientations are also described by simple PS relaxation models in section 3.3.2. A similar picture emerges from the MD study presented in section 3.3.3, provided that spurious spatial correlations are removed with supercells containing only an even number of cubic cells (Fig. 5). A random isotropic molecular tumbling is described by a single relaxation time in ref. 60, whereas a more elaborate isotropic model leads to the definition of a relaxation time for each angular momentum in ref. 61. The PS and rotator models presented in this work explicitly take into account both the symmetry of the cubic cage and the vectorial symmetry related to the C–N axis. This idea is close to the jump model proposed in ref. 61, where the molecular tumbling is described by a C_{4} symmetry in the tetragonal phase, neglecting inversion and mirror reflections. The same symmetry was used in the cubic phase, although a more correct approach should require the O point group as quoted by the authors. The two models of sections 3.3.2 and 3.3.3 correspond to the O_{h} point group, essentially adding the inversion symmetry (O_{h} = O × C_{i}) to the description of ref. 61. This is necessary to account for both electric and elastic multipolar effects, as well as to be able to discuss the possibility of ferroelectric phase transitions. Further analyses of neutron scattering data using symmetry adapted O_{h} rotators should provide a better description of the rotational molecular disorder in HOP.
Fig. 7 Schematic representation of possible ferroelectric phase transitions in HOP from the Pmm cubic phase (Table 4). These transitions involve various components of the Γ_{4}^{−} degenerate IR. MA cations are schematically represented by electric dipoles. |
Order parameters | Γ _{4} ^{−} | Z | Example | ||
---|---|---|---|---|---|
q_{1} | q_{2} | q_{3} | |||
P4mm | 0 | 0 | a | 1 | MAPbI_{3} |
Amm2 | a | a | 0 | 2 | FASnI_{3} |
R3m | a | a | a | 3 | MAGeCl_{3} |
For MAPbI_{3}, the P4mm polar group is a possible alternative to the cubic Pmm space group for structural refinement of the high temperature diffraction data.^{44} Our MD simulations however suggest that local polar symmetry breaking may only occur in relation to frozen configurations of the cations, and not collective fluctuations. The Amm2 appears in the crystallographic analysis of the room temperature phase of FASnI_{3}.^{44} The R3m space group is relevant for germanium compounds, MAGeX_{3}.^{43,76} As shown by a systematic study on germanium iodide compounds, CsGeI_{3}, MAGeI_{3} and FAGeI_{3},^{76} the appearance of a ferroelectric trigonal R3m space group and thus a large SHG response was attributed to the high density of states in the valence band due to sp-hybridization of the Ge and I orbitals. It shows that ferroelectricity for this class of HOP and AIP is essentially related to the instability of the Pmm cubic structure against a phonon mode related to the inorganic lattice, rather than collective reorientational motions of the cations. However, for larger organic cations like GA, DMA or TMA, the influence of the lattice instability is reduced and the size and shape of the cations may induce a variety of crystallographic structures, some of them being non-polar like P2_{1}/c for GAGeI_{3}.^{76}
Another possible outcome of linear translation–rotation couplings, even more characteristic of the stochastic molecular dynamics, is an additional acoustic damping related to the tumbling relaxational dynamics.^{20} When combining a Lagrangian, which includes the total free energy plus an additional kinetic energy term depending on the density ρ, with a dissipative part accounting for frictional forces, it is possible to predict the relaxation dynamics of the tumbling as well as its effect on acoustic wave propagation.^{20,79} Keeping only a damping matrix Γ for the relaxational motion, the set of linearly coupled dynamical equations reads:
For a single orientational degree of freedom, the relaxation rate without translation–rotation coupling is: γ = 1/τ = [χ_{0}^{−1} + J + j]/Γ. Separating the two degrees of freedom leads to a dressed dynamical matrix , where the Laplace transform variable is z = ω + iε^{+} in the limit of vanishing bare phonon damping. This result is similar to the one discussed above (section 3.2) for the coupling between a relaxor and optical modes in KMnF_{3} or SrTiO_{3}, but the impact of damping renormalization is expected to be very important and close to the Γ point. The coupling will thus lead to two separate signatures, one from relaxor and another from the acoustic phonon, as long as the acoustic phonon energy is much larger than the relaxation energy. However, due to acoustic phonon dispersion, a strong damping affects the whole acoustic phonon branch when k is close to the BZ center. Such a strong damping may affect a large part of the acoustic phonon spectrum in hybrid perovskites, especially the transverse acoustic phonon branches that are limited to about 2 meV over the entire BZ (section 3.2), given that the typical relaxation time of 5 ps (Fig. 6) corresponds to 0.12 meV. Thus, a careful investigation of the temperature dependence of the acoustic phonon spectrum would yield fruitful information about the influence of the stochastic dynamics on the translation–rotation coupling.
Some of the related antiferro-distortive space group changes associated with order parameters for AIP (P4/mbm) and HOP (I4/mcm and Pnma) are summarized in Table 5 and illustrated in Fig. 8 and 9. M_{3}^{+} and R_{4}^{+} define octahedra tilting, whereas Γ_{3}^{+}, M_{2}^{+}, and R_{3}^{+} correspond to the secondary order parameters.
Order parameters | M _{3} ^{+} | R _{4} ^{+} | ||||
---|---|---|---|---|---|---|
q_{1} | q_{2} | q_{3} | q_{4} | q_{5} | q_{6} | |
I4/mcm | 0 | 0 | 0 | a | 0 | 0 |
Pnma | 0 | a | 0 | b | 0 | b |
P4/mbm | a | 0 | 0 | 0 | 0 | 0 |
In oxide perovskites, Pmm to Pnma and Pmm to I4/mcm instabilities compete in the same way as in MAPbI_{3} and MAPbBr_{3}. The Pmm to Pnma instability is related to the simultaneous condensation of R_{4}^{+} and M_{3}^{+} rotational modes of the octahedra. Our symmetry analysis confirms that an M_{3}^{+} rotational mode, mainly related to the inorganic lattice, exists in the high temperature phase of HOP (Table 1). The Pmm to Pnma structural instability is indeed directly observed in the [536–602 K] temperature interval for CsPbI_{3}, leading to a Pnma phase at room temperature without the intermediate I4/mcm phase.^{41} In MAPbX_{3}, additional PS order parameters with the M_{3}^{+} symmetry exist for disordered collective molecular tumbling in scenarios A and B, as well as in the mixed scenario D (Table 1). They are expected to present an order–disorder character and to be linearly coupled with the displacive order parameter.
Finally, it appears that the Pm3m to P4/mbm instability is only observed in some AIP, namely CsPbCl_{3}.^{51} This instability is triggered by a heavily over-damped M_{3}^{+} rotational displacive mode of the lead halide octahedra in AIP. This result suggests that the potential anharmonicity associated to this normal vibrational mode is large in the cubic phase, and that the phase transition has an intermediate character between displacive and order–disorder. The Pm3m to P4/mbm phase transition is followed by P4/mbm to Cmcm and Cmcm to P2_{1}/m phase transitions (Fig. 9).^{51}
Fig. 10 (a) Atomistic representation at T = 400 K of the 444 supercell restricted to the CH_{3}NH_{3}^{+} cations. (Orange) cations having a very short relaxation time (τ < 1 ps), (green) cations with average relaxation time, (white) cations almost frozen on the time scale of the simulation (τ > 14 ps). (b) Time dependence of the autocorrelation function for all molecules within the 444 supercell extracted from MD simulation at T = 400 K. The average time dependence (Fig. 5 and 6) is plotted in bold dark for comparison. |
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