Mechanisms for collective inversion-symmetry breaking in dabconium perovskite ferroelectrics

Dabconium hybrid perovskites include a number of recently-discovered ferroelectric phases with large spontaneous polarisations. The origin of ferroelectric response has been rationalised in general terms in the context of hydrogen bonding, covalency, and strain coupling. Here we use a combination of simple theory, Monte Carlo simulations, and density functional theory calculations to assess the ability of these microscopic ingredients—together with the always-present through-space dipolar coupling—to account for the emergence of polarisation in these particular systems whilst not in other hybrid perovskites. Our key result is that the combination of A-site polarity, preferred orientation along 〈111〉 directions, and ferroelastic strain coupling drives precisely the ferroelectric transition observed experimentally. We rationalise the absence of polarisation in many hybrid perovskites, and arrive at a set of design rules for generating FE examples beyond the dabconium family alone.


Monte Carlo simulations Methodology
Metropolis Monte Carlo (MC) simulations were carried out using custom code based on that employed in Ref. S1 The fundamental degrees of freedom in all simulations were the MDABCO orientations, which were treated as classical unit spin vectors S i arranged on a simple cubic lattice. The S i were nearly always treated as Potts states; i.e. only a fixed number of possible orientations were possible for a given model-usually the eight possible 111 vectors. We discuss below the implications of using Heisenberg degrees of freedom with strong single-ion anisotropy. In general we used simulation boxes containing an 8 × 8 × 8 supercell of the primitive aristotypic cell (i.e. 512 'spins') and periodic boundary conditions were applied. Simulations were repeated in independent multiples of five. Equilibration times were estimated based on the number of MC steps required for the autocorrelation function to vanish within a specified (small) limit. For a given MC temperature point, each simulation was allowed to run for ten times as many moves as the corresponding equilibration time, and thermodynamic values were averaged over five successive collection runs, each spaced by this same number of MC steps. This means that the data points shown in Fig. 4(a) of the main text, for example, were each obtained as the average over 25 independent MC configurations.
MC energies were calculated using various combinations of the following various terms described in the text: The dipolar term was calculated using Ewald summation, giving a maximum relative error of 3.8×10 −5 .
Our implementation follows that of Ref. S1, which in turn is based on the implementations in Refs. S2-4. Two types of MC simulations were carried out. On the one hand, we sought sometimes to establish the ground state for a given interaction model. In such cases we used a simulated annealing approach as appropriate. On the other hand, we wished to determine the temperature dependence of other interaction models-specifically identifying the existence and nature of order/disorder phase transitions. In such cases we started our MC simulations at a temperature three times that of the dominant interaction term energy, and cooled at a relative rate of either 5% or 3% between successive MC temperature steps. A summary of the various MC simulations and their output is given in Table S1.

Effect of single-ion anisotropy
We make the point in the main text that allowing some deviation from 111 orientations in MC simulations of the key dipole-dipole + strain model preserves the paraelectric/ferroelectric phase transition but lowers the corresponding transition temperature. In support of this statement we report here the results of a MC simulation with anisotropic Heisenberg degrees of freedom. Note that the term E aniso [Eq. (4)] is minimised for S i ∈ 1 Degrees of freedom MC Energy Type Result Anisotropic Heisenberg Anisotropic Heisenberg Table S1: Summary of MC simulations carried out as part of this study. The abbreviations 'G.
S.' and 'T. D.' denote ground-state determination and temperature dependence MC simulation types, respectively.

Figure S1
: Temperature-dependent polarisation for the anisotropic Heisenberg MC models with strain and dipole-dipole interactions. The 8-state Potts trace is that shown in Fig. 4(a) of the main text. Evident here is that the same paraelectric/ferroelectric transition is observed for Heisenberg models with single-ion anisotropies Θ = 5, 10J, albeit with a lower transition temperature. Error bars are smaller than the symbols.

Dipole-dipole and strain calculations
Coarse-grained ground state energies for Models 1-5 [Eq. (4) of the main text] were calculated using Eqs. (2) and (3). The analytical forms of the corresponding energies and their contributions from dipoledipole and strain terms are collectively summarised in Table S2. Included in this table are the (fitted) coarse-grained and DFT energies used to construct Fig. 5
We used a plane wave basis set with a 800 eV energy cutoff and a 3 × 3 × 3 Monkhorst-Pack k-point mesh for the R3 structure (scaled accordingly for other supercells). Structures have been relaxed until the forces on any ion were less than 5 meV/Å.

Relaxed structures
The crystallographic details associated with the DFT-relaxed structures for Models 1-5 are given in Tables S3-S7. The atom labels used for MDABCO molecules are shown in Fig. S2. Atoms related to one another in the C 3v -symmetric MDABCO molecule but no longer symmetry-related in the corresponding ABX 3 polymorph are denoted by appending a suffix of the form 'a', 'b', or 'c'.

Symmetry implications of (anti)ferroelastic distortions
In the main text we discuss the implications of ferroelastic and antiferroelastic strains for distortions of the B-site coordination environment. We elaborate on the point here by making clear our symmetry arguments.
In the polar ferroelastic R3 phase, the B-site cation is located on the 3a Wyckoff site, with 3. point symmetry. The corresponding C 3 axis lies normal to one pair of faces of the RbI 3 octahedron. Hence the R3 state allows any distortion of the RbI 3 coordination environment that preserves this three-fold symmetry. This includes, in particular, rotations and anti-rotations (distortions towards a trigonal prismatic geometry) of the polyhedron that give bending both of Rb-I-Rb and I-Rb-I bond angles.
By contrast, the competing antiferroelastic phase has I23 space group symmetry, and the B-site cations are located on the 2a and 6b Wyckoff positions. These have 23. and 222.. point symmetry, respectively. In both cases, two-fold rotation axes pass through each of the Rb-I bond vectors, which constrains the corresponding Rb-I-Rb angles to be 180 • and the I-Rb-I angles to be 90 • . Hence there are no symmetry-allowed polyhedral rotations or bending modes in this state. We anticipate this raises the energy of the antiferroelastic phase, which is why the strain coupling strength J is positive, rather than negative, for [MDABCO]RbI 3 .