Simulation of Structural Phase Transitions in Perovskite

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C: Physical Processes in Nanomaterials and Nanostructures

Simulation of Structural Phase Transitions in Perovskite Methylhydrazinium Metal-Formate Frameworks: Coupled Ising and Potts Models Mantas Šim#nas, Andrius Ibenskas, Alessandro Stroppa, Anna G#gor, Miros#aw M#czka, Juras Banys, and Evaldas E. Tornau J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b03448 • Publication Date (Web): 21 Jul 2019 Downloaded from pubs.acs.org on July 23, 2019

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The Journal of Physical Chemistry

Simulation of Structural Phase Transitions in Perovskite Methylhydrazinium Metal-Formate Frameworks: Coupled Ising and Potts Models Mantas Šim˙enas,∗,† Andrius Ibenskas,‡ Alessandro Stroppa,¶ Anna Gągor,§ Mirosław Mączka,§ J¯uras Banys,† and Evaldas E. Tornau‡ Faculty of Physics, Vilnius University, Sauletekio av. 9, LT-10222 Vilnius, Lithuania, Semiconductor Physics Institute, Center for Physical Sciences and Technology, Sauletekio av. 3, LT-10257 Vilnius, Lithuania, CNR-SPIN, c/o Dip.to di Scienze Fisiche e Chimiche - Università degli Studi dell’Aquila - Via Vetoio - 67100 - Coppito (AQ), Italy, and Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box-1410, PL-50-950 Wroclaw 2, Poland E-mail: mantas.simenas@ff.vu.lt Phone: +370 5 2234537. Fax: +370 5 2234537

∗

To whom correspondence should be addressed Faculty of Physics, Vilnius University, Sauletekio av. 9, LT-10222 Vilnius, Lithuania ‡ Semiconductor Physics Institute, Center for Physical Sciences and Technology, Sauletekio av. 3, LT10257 Vilnius, Lithuania ¶ CNR-SPIN, c/o Dip.to di Scienze Fisiche e Chimiche - Università degli Studi dell’Aquila - Via Vetoio 67100 - Coppito (AQ), Italy § Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box-1410, PL-50-950 Wroclaw 2, Poland †

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Abstract We present a theoretical model of the methylhydrazinium CH3 NH2 NH2+ molecular cation ordering in perovskite [CH3 NH2 NH2 ][M(HCOO)3 ] (M = Mn, Mg, Fe and Zn) formate frameworks, which exhibit two structural phase transitions. The proposed model is constructed by analyzing the available structural information and mapping the molecular cation states on a three-dimensional simple cubic lattice. The model includes the short-range Ising and Potts interactions between the dipolar CH3 NH2 NH2+ cations. We study the model using the Monte Carlo simulations as well as by the Density Functional Theory calculations. The simulations indicate that our model accurately describes the methylhydrazinium cation arrangement in all three structural phases of the compounds. The calculated temperature dependences of the heat capacity and electric polarization are in good agreement with the experimental data. The simulations also allow us to obtain the characteristic energies of the molecular cation interactions for all members of the [CH3 NH2 NH2 ][M(HCOO)3 ] family.

Introduction Over the past decade, hybrid [A][M(HCOO)3 ] metal-formate frameworks have received a significant attention from the scientific community due to the possibility of ferroelectric 1,2 and multiferroic 3–5 behavior. In these compounds, A+ denotes a molecular alkylammonium cation (e.g. (CH3 )2 NH2+ ), and M2+ is a divalent transition metal or Mg2+ ion. 6,7 The metal ions are joined together by the anionic formate HCOO– linkers thus forming structures with nanocavities. Each such cavity contains a single molecular cation, which is H-bonded with the metal-formate framework. 8 The majority of formate frameworks exhibit structural phase transitions related to the cooperative molecular cation ordering and framework deformation. 9–11 Some of these transitions are accompanied by the appearance of the spontaneous electric polarization 12,13 and ferroelectric hysteresis. 1,2,14 The compounds with magnetic transition metal ions also exhibit long-range magnetic order 4,5,11,15–17 making these 2


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materials potential single phase multiferroics. 18 The properties of the phase transitions in formate frameworks highly depend on the metal center and type of the molecular cation. 6 So far the most thoroughly studied family of this class of materials is [(CH3 )2 NH2 ][M(HCOO)3 ] (DMAM), which contains dimethylammonium (DMA+ ) molecular cations. 3,9,15 DMAM frameworks crystallize in a perovskite topology and show a single temperature-induced structural phase transition to a phase with the spontaneous electric polarization. 2,19,20 Other interesting members are chiral ammonium metal-formates [NH4 ][M(HCOO)3 ], which exhibit clear ferroelectric switching behavior. 1,4 However, the structural phase transitions in these and most of the other formates occur well below the room temperature obscuring their potential applicability. This stimulates the search and characterization of new formate frameworks with spontaneous electric polarization at room temperature. One family of such compounds with polarization existing above 300 K is methylhydrazinium (MHy+ ) metal-formate frameworks [CH3 NH2 NH2 ][M(HCOO)3 ] (MHyM, where M = Mn, Mg, Fe and Zn). 11 Recently, these materials were investigated using various experimental techniques, which revealed two structural phase transitions. 11–13 Depending on a metal center, the temperatures of the transitions are in the range Tc1 = 310 − 327 and Tc2 = 168 − 243 K. Similarly to DMAM frameworks, the structure of MHyM compounds has the perovskite topology, where the polar MHy+ cations are embedded in the cuboid metal-formate cavities (see Figure 1). In the high-temperature (HT) phase (Tc1 < T ), each MHy+ cation can hop around the three-fold and flip around the two-fold axes resulting in six equivalent positions (Figure 1a). The three-fold axis goes through the terminating carbon and nitrogen atoms and it is roughly perpendicular to the two-fold axis. In the intermediate (I) phase (Tc2 < T < Tc1 ), the flipping motion is absent, and all MHy+ cations are pointing towards the same direction (Figure 1b). This results in the appearance of the spontaneous electric polarization at Tc1 as demonstrated by the pyrocurrent measurements. 11–13 The hopping motion ceases in the low-temperature (LT) phase (T < Tc2 ), where all MHy+ cations 3



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establish the long-range checkerboard order (Figure 1c). The phase transition to the LT phase is accompanied by a further increase of the electric polarization. 11–13 (b)

(a)

(c) Mn N

C2

C N/C

C3

O

C3

H HT phase Tc1 < T 6-fold disorder

I phase Tc2 < T < Tc1 3-fold disorder

LT phase T < Tc2 Ordered

Figure 1: Crystal structure of MHyMn framework in the (a) HT, (b) I and (c) LT phases. The hydrogen atoms of the MHy+ cations in the HT and I phases are not shown for clarity. C3 and C2 mark the three-fold and the two-fold rotation axes, respectively. Structural data taken from Ref. 11 Despite thorough experimental investigations, currently there are only few theoretical studies of hybrid frameworks, 20–23 and still no theoretical description of the phase transitions in MHyM family. Recently, we proposed and simulated using the Monte Carlo (MC) approach a phase transition model, which successfully describes the DMA+ cation ordering in DMAM frameworks. 20 We demonstrated that this model can be mapped to the classical 3-state Potts model, 24 while a better agreement with the experimental data was obtained by considering additional long-range dipolar interactions. Another recent MC study by Coates et. al. of cyanoelpasolite frameworks revealed that the structural phase transitions in these compounds can also be well described by the Potts models. 23 The main advantage of the MC simulations lies in the capacity to exploit large multiparticle systems and entropic effects, which are usually not accessible by the Density Functional Theory (DFT) calculations. In this study, we significantly extend our previous DMAM model to describe both structural phase transitions in MHyM formate frameworks. In the modified model, the phase transition from the HT to I phase is essentially described by the classical Ising model, while 4


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the transition to the LT phase retains the features of the 3-state Potts model. The model is studied using the MC simulations assisted by the DFT calculations. Our numerical simulations correctly mimic the experimentally observed MHy+ cation ordering in all members of the MHyM formate family and reveal the appearance of the spontaneous electric polarization at both phase transition points.

Model and Simulation Details A microscopic model describing the phase transitions in MHyM formates requires a detailed analysis of the available structural data. A complete crystal structure in all three phases is only known for the Mn and Zn members of the family. 11,13 However, all compounds are expected to be isostructural and to share the same phase transition mechanism. Thus, for the structural data analysis we have chosen MHyMn compound. A larger structural motif of MHyMn framework in the HT and LT phases is presented in Figure 2 revealing two types of cavities with differently oriented MHy+ cations. We refer to these cavities as A and B. In the HT phase, each type of cavity accommodates six possible MHy+ states (orientations), which occur due to the three-fold hopping and two-fold flipping dynamics of the cations. However, the six states in both cavities are not the same, and for this reason the MHy+ cations in the LT phase form a three-dimensional checkerboard arrangement, where each cation has the nearest neighbors (NN) of different orientation (Figure 2b). Thus, in total there are twelve possible cation states, which are depicted in Figure 3. States S = {1, 2, 3, 4, 5, 6} can only occur in the cavities of type A, while S = {7, 8, 9, 10, 11, 12} are contained in the B-type cages. In addition, states S = {1, 2, 3, 7, 8, 9} have opposite orientation with respect to the position of the terminal carbon atom as compared with the other six states. The basis of our model is the mapping of the experimental framework structure on a simple cubic lattice, where each lattice point corresponds to a single MHy+ cation (see

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eI and eP interactions A

B

B

B

A

B

A

A

A

B

B

A z y

(b)

(a)

x

(c)

Figure 2: Structural motif of MHyMn framework in the (a) HT and (b) LT phases. Two types of cavities (sublattices) are marked as A and B. (c) Mapping of the experimental LT structure on the simple cubic lattice with Ising and Potts type NN interactions (red arrows) between the MHy+ cations.

S=2

S=3

S=7

S=8

S=9

S=4

S=5

S=6

S = 10

S = 11

S = 12

ψ = +1 Ising states

S=1

ψ = −1 Ising states

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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B-type cavity

A-type cavity

Figure 3: Model states of the MHy+ cations. The Potts-type interactions occur only between the following states from the neighboring sublattices: (1, 7), (2, 8), (3, 9), (4, 10), (5, 11) and (6, 12). States S = {1, 2, 3, 7, 8, 9} correspond to the Ising variable σ = +1, while σ = −1 is assigned to S = {4, 5, 6, 10, 11, 12}.

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Figure 2c). The mapping maintains both types of cavities, which in the model are called A and B sublattices. We consider interactions only between the NN cation states, and do not explicitly take into account the M(HCOO)3– framework, though its deformation induced by the H-bonds acts as a medium for transfer of the interactions. Note that such simplified effective models were demonstrated to correctly capture the cation ordering and phase transition sequences in similar hybrid compounds. 20,25,26 The interactions between the cations are also determined by analyzing the preferred cation arrangement in the I and LT phases. During the phase transition from the HT to I phase, the flipping motion of the MHy+ cations is reduced, and the terminal carbon atoms tend to point approximately to the same direction. This situation resembles the ordering behavior of the famous Ising model, and hence we select it as a candidate to describe the phase transition at Tc1 . As demonstrated in our previous study of DMAM family, 20 the framework deformation in the LT phase strictly limits only certain molecular cation arrangements, and this ordering can be modelled with the Potts model, which was initially used to describe the interactions between the spins and magnetic phase transitions. 24 Our model Hamiltonian is a sum of the Ising and Potts-type interactions:

H = HI + HP .

(1)

The Ising Hamiltonian is X

HI = −eI

ψi ψj ,

(2)

where the Ising state variable ψ = +1 corresponds to S = {1, 2, 3, 7, 8, 9}, and ψ = −1 is assigned to S = {4, 5, 6, 10, 11, 12} cation states (see Figure 3). The indices i and j enumerate the sites on the simple cubic lattice. The summation excludes double counting and takes into account only the NN sites. The strength of this interaction is determined by the positive energy parameter eI . Such Ising representation ensures that two neighboring cations will have lower energy, if their terminal carbon atoms are pointing towards the same direction. 7



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The arrangements where these atoms are oriented in opposite direction will have higher energy. For example, neighboring cation states 1 and 7 (both ψ = +1) will be preferred as their interaction energy corresponds to −eI , while states 1 and 10 (ψ = +1 and ψ = −1, respectively) will result in a much higher energy of +eI . Thus, the ground state of this Hamiltonian will consist of either ψ = +1 subset or ψ = −1 subset of the cation states. The explicit form of the Potts interaction responsible for the lower temperature phase transition is:

HP = −eP

X

[δ(Si , 1)δ(Sj , 7) + δ(Si , 2)δ(Sj , 8) + δ(Si , 3)δ(Sj , 9)+

+ δ(Si , 4)δ(Sj , 10) + δ(Si , 5)δ(Sj , 11) + δ(Si , 6)δ(Sj , 12)]. (3) Here δ(m, n) is the Kronecker delta, which is equal to one for m = n and zero otherwise. This expression ensures that the interaction can only occur between the following neighboring states: (Si , Sj ) = (1, 7), (2, 8), (3, 9), (4, 10), (5, 11) and (6, 12). The energy of this interaction is equal to −eP and zero for all other geometrically possible pairs. The ground state of this Hamiltonian is expected to match the experimentally observed checkerboard arrangement (Figure 2b) 11,13 spanned by one of the six preferred pairs of the cation orientations. 20 In our previous study of the phase transition in DMAM frameworks, 20 we demonstrated that a similar Potts-type Hamiltonian with only six DMA+ cation states can be mapped to the 3-state Potts model. 24 Deep in the I phase, the current model also has only six MHy+ states, as the Ising-type interactions described by HI eliminate either S = {1, 2, 3, 7, 8, 9} (ψ = +1) or S = {4, 5, 6, 10, 11, 12} (ψ = −1) subset of the cation states. Thus, the same mapping procedure can be applied to the Hamiltonian HP in the I phase revealing its analogy to the 3-state Potts model given by: 24

H3P = −eP

X

δ(σi , σj ).

8


(4)

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Here σ = {1, 2, 3} denotes the Potts variable. For ψ = +1 ground state of the Ising Hamiltonian, σ = 1 for S = {1, 7}, σ = 2 for S = {2, 8} and σ = 3 for S = {3, 9}, while for ψ = −1, σ = 1 for S = {4, 10}, σ = 2 for S = {5, 11} and σ = 3 for S = {6, 12}. Note that in our previous studies of DMAM and related compounds, 20,25 we also included the dipolar interactions between the molecular cations. Such long-range interactions could influence the ordered structures, change the order of the phase transition and generate structural twins, 20,25,27–30 though they lead to the significantly more complicated simulations. Our current model already involves twice as many states as used to describe the ordering in DMAM, and thus for simplicity we do not consider dipolar interactions between the MHy+ cations. Still we remain aware that the dipolar effects are possible in this system. We estimated the value of the Ising interaction energy eI using the DFT calculations of all allowed mutual MHy+ cation arrangements. To produce these geometries, we had to reduce the symmetry of the reported rhombohedral HT phase 11 to the triclinic P 1 using [2/3 1/3 1/3 −1/3 1/3 1/3 −1/3 −2/3 1/3] transformation matrix and Jana2006 transformation tools. 31 The generated triclinic unit cell (lattice parameters: a = b = c = 9.016(1) Å, α = β = γ = 54.21(1)◦ and volume V = 448.2 Å3 ) accommodates all atoms at the general positions allowing to obtain all 36 possible combinations between the two MHy+ cations. Half of these combinations correspond to the cations oriented in the same direction (−eI energy), while the other half point oppositely (+eI ). DFT calculations were performed using the Vienna ab-initio Simulation Package 32 using the projector augmented wave (PAW) potentials within the PBE approximation to exchange-correlation functional. 33 The cutoffenergy was fixed to 600 eV with a 3×3×3 k-point mesh. During the calculations, the hydrogen atoms were allowed to relax, while other atoms were fixed in the positions determined by the XRD. These preliminary calculations of the total energy of the different cation configurations provided an initial estimation of eI of 9.5 meV or 110 K, which we used as a starting value in our simulations. Subsequently both eI and eP parameters were varied to fit the experimental data.

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The model Hamiltonian was solved using the MC simulations based on the single-flip Metropolis algorithm. 34 All calculations started from a randomly chosen lattice configuration. First, a point in the lattice was randomly selected and its initial interaction energy Ei was evaluated using Eq. 1. Afterwards the chosen state was randomly changed to one of the remaining states, and the final energy Ef was calculated. The new state was accepted with a probability min (1, e−∆E/kB T ), where ∆E = Ef − Ei is the energy difference, kB denotes the Boltzmann constant, and T is the temperature. Regardless of the outcome, a new lattice site was randomly selected, and the Metropolis procedure was repeated. The MC simulations were performed on a simple cubic lattice with the implemented periodic boundary conditions. The lattice size was L × L × L, where L = 10 − 20 (in terms of the lattice constant). At each temperature, we discarded initial 105 MC steps and then used up to 5 × 106 steps for calculation of the thermodynamic averages. To probe the phase transition properties of our model, we calculated the heat capacity at constant volume using the following equation:

CV =

< H2 > −< H >2 , kB T 2 V

(5)

where V = L3 is the volume of the system, and the angle brackets denote the MC average. The normalized electric polarization vector P~ = (Px , Py , Pz ) was obtained by averaging the total electric dipole moment of the cation system: < P~ =

P

p~i > . p0 V i

(6)

Here p~i = p0 pî is a dipole moment of a cation state at lattice site i. We assume that all cation states have the same dipolar magnitude p0 , but different direction pˆ. To determine the unit vector pˆ of each cation state, we first performed a DFT estimation of pˆ for the isolated MHy+ cation. This calculation indicated that the dipole moment is pointing from the middle of the C−N bond to the terminal nitrogen atom. We used this result together 10


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with the available structural data 11 to assign the unit vectors to all twelve MHy+ cation states.

Simulation Results and Discussion We investigated the proposed phase transition model of MHyM frameworks using the MC simulations. The snapshots of the simulated HT, I and LT phases in the cation state representation are shown in Figure 4a. The Ising ψ and Potts σ variable representations of the lattice are provided in Figure 4b and c, respectively. All MHy+ cation states are present in the HT phase (350 K) indicating disordered system. The disorder is also revealed in the ψ and σ representations of the lattice, though some shortrange clustering of the states can be identified. Note that some molecular cation correlations in the disordered phases of formate frameworks were also observed experimentally. 35,36 In our model they simply occur due to the expected increase of the correlation length on approaching the phase transition temperature. 34 In the I phase (200 K), a partial cation order is established, since half of the states are no longer available due to the Ising phase transition at Tc1 . In the ψ-representation, the lattice is completely ordered, as almost all available states belong to the ψ = +1 subset, while the Potts representation still indicates disorder. In the LT phase (100 K), the complete long-range MHy+ cation order is observed in all representations, and the obtained ground state corresponds to the expected checkerboard pattern of the cations. Note that due to the inherent model symmetry, there are six equivalent ground states with different cation orientations. The obtained MHy+ cation arrangement in all structural phases is in a perfect agreement with the experimental data. 11–13 We further calculated the temperature dependence of the heat capacity of our model (see Figure 5). To fit the experimentally observed phase transition temperatures of MHyM compounds, we used different values of eP and eI interaction parameters summarized in Table 1. The eP parameter significantly decreases on going from Mg to Zn metal center, while eI

11



Figure 4: (a) Snapshots of the MC simulations of our model at different temperature corresponding to the HT, I and LT phases of MHyM framework. The MHy+ cation states are color-coded. The same snapshots represented using the (b) Ising ψ and (c) Potts σ variables. Simulation parameters: L = 20, eP = 93 and eI = 55 K.

12


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shows a slight increase. The determined eP /eI ratio varies from 1.7 to 3.0 for MHyZn and MHyMg frameworks, respectively. The observed differences of this ratio likely originate from the different ionic character of the M−O bonds. 11,37 Note that the determined eI parameter values are roughly two times smaller than predicted by our DFT estimation. A likely source of this discrepancy is that our model cannot take into account such a detailed interaction picture as DFT. In addition, our DFT calculations are based on the HT crystal structure of MHyMn with a slightly reduced symmetry, which may also contribute to this difference. The simulated temperature dependence of CV is very similar to the experimental data (compare Figure 5a and b). In both cases the peak of the heat capacity at Tc1 is significantly smaller and broader compared with the anomaly at Tc2 . The Ising model exhibits a second-order (continuous) phase transition, which is in a perfect agreement with the experimental data of MHyM frameworks. 11,13 The 3-state Potts model exhibits a weak first-order (discontinuous) phase transition. 24 Indeed the same character of the phase transition at Tc2 was also observed for MHyZn compound using magnetic resonance methods. 12 The accurate agreement with the experimental data indicates that our model correctly describes not only the cation arrangement in all structural phases, but also the temperature evolution of the ordering. Note that accounting for the long-range dipolar interactions between the molecular cations may further improve the agreement with the experimental data, as we recently demonstrated for DMAM and lead-halide hybrid perovskites. 20,25 Table 1: Experimentally determined phase transition temperatures of MHyM frameworks 11 and the respective values of the Ising and Potts type interaction parameters obtained from the MC simulations. Framework MHyMg MHyMn MHyFe MHyZn

Tc1 (K) Tc2 (K) eI (K) eP (K) 327 243 46 138 310 224 45 127 310 183 50 102 324 168 55 93

eP /eI 3.0 2.8 2.0 1.7

We further simulated the heat capacity of the pure Ising and 3-state Potts models to verify the equivalence with our model. The calculations were performed by fixing the same 13



40 (a) MC MHyMg

CV /kB

30

MHyMn

20

MHyFe MHyZn

10

ΔCp (J·mol−1·K−1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0 120

(b) Exp.

90 60

Tc1

Tc2

30 0

150

200 250 300 Temperature T (K)

350

Figure 5: (a) Simulated temperature dependence of the heat capacity of our model for different MHyM frameworks and (b) the corresponding experimental data previously reported in Ref. 11 Simulation parameters are summarized in Table 1, and L = 20. energy parameters eP and eI as obtained for MHyZn compound. The obtained results are presented in Figure 6 revealing a perfect correspondence between the models with respect to the shape of the CV curves. The phase transition temperature Tc2 of our model matches that of the pure Potts model, while Tc1 is higher compared to the Ising model. This indicates that the Potts interactions also affect the Tc1 phase transition temperature. Note that Tc1 of our model determined for eP = 0 is in a perfect agreement with the pure Ising model (not shown). We also calculated the temperature dependence of the electric polarization Pz along the z-axis (reference frame is defined in Figure 2c) for MHyZn framework (see Figure 7). In the HT phase, Pz is zero due to random MHy+ cation dynamics, while at both phase transition points the polarization spontaneously increases indicating pyroelectric behavior. A larger increment at Tc1 is related to the end of the flipping motion followed by the alignment of all dipole moments towards the same direction. The cease of the hopping motion at Tc2 causes 14


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CV /kB

30 MHyZn

20

3-state Potts Ising

10 0

150

200 250 300 Temperature T (K)

350

Figure 6: Comparison of the temperature dependent heat capacities of our model with pure Ising and 3-state Potts models. a much smaller increase of Pz . The result of this simulation is in a good agreement with the experimental polarization measurements. 12,13

1.0

MHyZn

Polarization Pz

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.8 0.6 0.4

Tc1 Tc2

0.2 0 50 100 150 200 250 300 350 Temperature T (K)

Figure 7: Temperature dependence of the z-component of the normalized polarization of MHyZn framework obtained by MC simulations. Simulation parameters: L = 20, eP = 93 and eI = 55 K. The proposed phase transition model is essentially a 6-state model, since six cation states are available in each cavity at high temperature. Thus, the total entropy change of our model during both transitions is ∆S = ∆S1 + ∆S2 = V kB ln 6, where ∆S1 = V kB ln N1 = V kB ln 2 and ∆S2 = V kB ln N2 = V kB ln 3. The changes of entropy obtained from the differential scanning calorimetry measurements correspond to the significantly smaller degrees of disorder: N1 = 1.3 and N2 = 1.4. 11 Note that the entropy was also overestimated in some 15



structural studies of DMAM frameworks, 3,9 while other investigations revealed no discrepancy. 38,39 The entropy problem observed for MHyM family might suggest a more complex ordering mechanism than described by our model, some experimental uncertainties or influence of the defects in the real compounds. To test the last hypothesis, we performed the MC simulations of our model with randomly distributed defects, which correspond to the non-interacting and non-rotating MHy+ cation sites. Such defects do not participate in the ordering, but can disrupt the long-range cation order. We found that to account for the observed entropy difference, the crystal lattice should contain more than 50% of defect sites making this scenario unlikely. At reasonable levels of defect concentration (≤ 1%), we observed only negligible effects on the phase transition properties of our model.

Summary and Conclusions In this study, guided by the available structural data, we constructed and numerically solved the phase transition model of the MHy+ cation ordering in MHyM formate frameworks. Our model takes into account the short-range Ising and Potts-type interactions between the neighboring molecular cations. The Ising part of the model describes the higher temperature phase transition, characterized by the termination of the flipping dynamics of the MHy+ cations and establishment of the partial long-range dipolar order. The Potts interactions are responsible for the lower temperature transition related to the cooperative freezing of the hopping cation motion and complete ordering of the system. Our model correctly reproduced the whole phase transition sequence and MHy+ cation arrangement in all structural phases of the MHyM compounds. It was further validated by the temperature dependences of the heat capacity and electric polarization, which were found to be in the perfect agreement with the experimental results. Our simulations also allowed us to deduce the characteristic Potts and Ising interaction energies between the molecular cations for different members of the MHyM family. The determined ratio of these energies

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varies in a qualitative agreement with the change of the ionic character of the M−O bond. We also addressed the problem of discrepancy of the entropy change in formate frameworks and demonstrated that it cannot originate from the defect sites of the molecular cations. Here we would also like to identify the main shortcomings of the model and their implications. Firstly, the HT phase of MHyM compounds has a trigonal crystal symmetry, while the mapping of the MHy+ cation subsystem is carried out on the simple cubic lattice. This may result in some differences between the experimental and simulated data, though we tried to minimize this effect by determining the dipole moments of the MHy+ cations directly from the structural data. Secondly, in our model the metal-formate framework is not explicitly taken into account and acts only as the interaction mediator. This might introduce some discrepancies in the calculation of the polarization, as M(HCOO)3– units are also expected to have an electric dipole moment. We also ignored the dipolar interactions between the molecular cations, which tend to change the phase transition order and introduce layered cation arrangements, as was recently demonstrated for DMAM frameworks. 20 Finally, we note that the Ising model can be considered as the 2-state Potts model. Hence, our model essentially combines two Potts models with different number of states. Recently, the 4- and 6-state Potts models were also used to describe the ordering in cyanoelpasolite [C3 N2 H5 ]2 Rb[Co(CN)6 ] and [NMe4 ]2 B[Co(CN)6 ] (B = K, Rb, Cs) hybrid perovskite compounds. 23 This suggests a general rule that any structural phase transition in hybrid frameworks can be described by the q-state Potts model, where q denotes the degree of disorder of a molecular cation.

Acknowledgement This work was supported by the Research Council of Lithuania (Project TAP LLT-4/2017).

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Simulation of Structural Phase Transitions in Perovskite

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