Exploring Antipolar Nature of Methylammonium Lead Halides: a

Letter pubs.acs.org/JPCL

Exploring the Antipolar Nature of Methylammonium Lead Halides: A Monte Carlo and Pyrocurrent Study † ‡ § Mantas Šimeṅ as,*,† Sergejus Balčiunas, Mirosław Mączka,‡ Juras ̅ ̅ Banys, and Evaldas E. Tornau †

Faculty of Physics, Vilnius University, Sauletekio 3, LT-10257 Vilnius, Lithuania Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box-1410, PL-50-950 Wroclaw 2, Poland § Semiconductor Physics Institute, Center for Physical Sciences and Technology, Sauletekio 3, LT-10257 Vilnius, Lithuania ‡

S Supporting Information *

ABSTRACT: The high power conversion efficiency of the hybrid CH3NH3PbX3 (where X = I, Br, Cl) solar cells is believed to be tightly related to the dynamics and arrangement of the methylammonium cations. In this Letter, we propose a statistical phase transition model which accurately describes the ordering of the CH3NH3+ cations and the whole phase transition sequence of the CH3NH3PbI3 perovskite. The model is based on the available structural information and involves the short-range strain-mediated and long-range dipolar interactions between the cations. It is solved using Monte Carlo simulations on a three-dimensional lattice allowing us to study the heat capacity and electric polarization of the CH3NH3+ cations. The temperature dependence of the polarization indicates the antiferroelectric nature of these perovskites. We support this result by performing pyrocurrent measurements of CH3NH3PbX3 (X = I, Br, Cl) single crystals. We also address the possible occurrence of the multidomain phase and the ordering entropy of our model.

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Among many different experimental studies, ordering and dynamics of the MA+ cations were investigated using theoretical methods such as density functional theory (DFT)16,24,29−35 and molecular dynamics (MD).26,31,36−38 The obvious advantage of DFT is that it can provide the atomistic details of the crystal structure.30 However, it is difficult to employ this method to study the dynamic and temperature-induced entropic effects (e.g., phase transitions and thermal fluctuations). The influence of temperature can be investigated using MD,38 but such simulations may suffer from the pronounced finite-size effects that could obscure the longrange order in the system. Another method of choice to study phase transitions is Monte Carlo (MC) simulations that can easily exploit much larger systems and entropic effects, but in this method the atomistic picture must be simplified to a coarse-grained model. Several MC studies dedicated for the MAPbX3 systems are already reported.25,33,39−42 Most of them concentrate on the dynamics of the MA+ cations,25,39,41,42 and only few briefly address one of the structural phase transitions.33,40 To our knowledge, a proper MC study, which would describe the whole sequence of phase transitions in these materials, is still absent. Thus, in this Letter we propose a statistical phase transition model which is based on the experimentally determined structural information on MAPbI3 perovskite and correctly describes the complete phase transition sequence of this

ecently, perovskite methylammonium lead halides CH3NH3PbX3 (where X = I, Br, Cl) have attracted exceptional attention of the scientific community due to their potential application as effective and affordable solar cells.1−3 The power conversion efficiency of cells based on these hybrid compounds was improved from several to more than 20% in less than a decade.4−9 Such a high efficiency of these materials is a consequence of several physical factors including a large absorption coefficient, long carrier diffusion lengths and suitable band gap.10−13 It is an established notion that the microscopic origin of the low carrier recombination is connected to the relatively unhindered motion of the methylammonium CH3NH3+ (MA+) cations, while the precise mechanism is still under intense debate.14−20 The movement of the MA+ cations is tightly related to the structural phase transitions in these materials. The number of these transitions and their temperatures depend on the type of the halide: two phase transitions are observed in iodide (at 327 and 162 K) and chloride (179 and 173 K), while the bromide analogue exhibits three transformations at 237, 155, and 145 K.21 During the phase transitions, the crystal symmetry changes from the highly symmetric cubic to the tetragonal and then to the orthorhombic. There are some indications that the latter phases might be ferroelectric,22,23 which may be the reason for the high carrier diffusion lengths and low recombination rates.15,19,24 However, attempts to obtain the electric polarization hysteresis loops were unsuccessful rising serious doubts about ferroelectricity and influence of the ferroelectric domains on the high efficiency of these perovskites.16−18 The possibility of the antiferroelectricity18,25,26 and ferroelastic27,28 domains is widely discussed in this context. © XXXX American Chemical Society

Received: August 24, 2017 Accepted: September 25, 2017 Published: September 25, 2017 4906

DOI: 10.1021/acs.jpclett.7b02239 J. Phys. Chem. Lett. 2017, 8, 4906−4911

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The Journal of Physical Chemistry Letters

Figure 1. Experimental structure of MAPbI3 perovskite in (a) cubic, (b) tetragonal and (c) orthorhombic phases (taken from ref 43). The PbI6 octahedra are indicated in (a), while the MA+ cations are presented as a superposition of several orientations. In most of the cubes the hydrogen atoms are omitted for clarity. The reference frame of the crystal axes is defined as in the cubic phase.

compound. We supplement the theoretical results by the electric pyrocurrent measurements of the MAPbI3, MAPbCl3, and MAPbBr3 single crystal samples. Both theory and experiment support the idea of the antipolar (antiferroelectric) nature of methylammonium lead halides. The structure of the MAPbI3 perovskite, which was used to construct the phase transition model,43 is presented in Figure 1. In the cubic phase, the MA+ cation is situated in the center of the cuboid cavity formed by the PbI6 octahedra (Figure 1a). The system is highly disordered, since the cation can easily rotate around the C−N axis and hop (reorientate) between six different positions. The resolved positions of the carbon atoms demonstrate that the C−N axes are pointing to the faces of the lead-iodide cubes. Four orientations of the MA+ cation also remain in the tetragonal phase indicating a partly ordered system, while the two states along the c-axis are absent (Figure 1b). In this phase, the cations are still exhibiting reorientation in the ab-plane.36,44 The long-range order is established in the orthorhombic phase, where the molecular cations are arranged in the checkerboard bidirectional manner in the ab-plane and exhibit an antiparallel (antiferroelectric) alignment along the caxis (see Figures 1c and S1 in the Supporting Information (SI)).43 Such a checkerboard arrangement of the MA+ cations was also obtained in several DFT and MD studies.16,26,29,31,35 Note that a very similar ordering sequence solely based on the symmetry arguments was already proposed in 1998 by Maalej et al.45 The presented structural information is used to construct our phase transition model of the MAPbI3 perovskite. Following the cation arrangement in the cubic phase, we introduce six pseudospin states S = {1, 2, 3, 4, 5, 6} of the MA+ that point to the faces of the cubes. The states are depicted as vectors that correspond to the electric dipole moments of the molecular cations (see Figure 2a). For simplicity, we consider that the direction of the dipole moment coincides with the C−N axis (vector is directed from carbon to nitrogen atom). The cubes represent the lead-iodide framework which we keep static and undeformed. Each such cube (or MA+ cation) is represented by a lattice point in our model. The experimentally observed deformation of the framework is taken into account indirectly

Figure 2. Model states of the MA+ cations and their interactions. (a) The mapping of the cation arrangement in the cubic phase to the six model states S = {1, 2, 3, 4, 5, 6}. (b) Two main model interactions between the states in the ab-plane. (c) A cross section of the threedimensional lattice of the orthorhombic phase of our model illustrating the cutoff radius of the sphere used to calculate the dipolar interactions.

by the strain-mediated interactions between the nearestneighbor (NN) cations. These interactions occur because each MA+ cation forms H-bonds with the iodine atoms that in turn affect the neighboring cells. Based on the structure of the orthorhombic phase,43 we select two main effective NN interactions between the cation states in the ab-plane. They correspond to the perpendicular arrangements of the cations with energies e1 and e2. The interaction e1 occurs if a vector is pointing to the center of a neighboring state (to the shared face of the two cubes). The cations interact with energy e2 for a vector pointing from the center of another state. Figure 2b shows an example of these interactions for states S = 1,4 (energy e1) and S = 1,2 (energy e2). We do not consider these interactions if they involve states S = 5 and 6 that are perpendicular to the ab-plane. Our choice of only two short-range interactions e1 and e2 is supported by a recent DFT study35 which revealed that they are dominant in the MAPbI3 perovskite. In our model e1 interaction is stronger than e2, since the amino group forms stronger H-bonds with the iodine atoms providing higher strain of the lattice.33,43 4907


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The Journal of Physical Chemistry Letters Another DFT investigation reported that a single MA+ cation possesses a substantial electric dipole moment of 2.29 D.14 Thus, the long-range dipole−dipole interactions between the cations are also expected to contribute to the ordering of the system. The Hamiltonian describing the dipolar interaction is /d = ed ∑

for the low-temperature phase transition. In contrast, the peak of CV at Tc1 is weakly affected by these interactions. The arrangement of the MA+ cations in different phases of our model are presented in Figure 4 and S3−4. There is no long-range order in the cubic phase, since the MA+ cations are constantly hopping between the six states. In the tetragonal phase, the states S = 5 and 6 are absent, since the cations are partially ordered into two sublattices of the ab-planes (see Figure S4). The first sublattice is randomly filled with the states S = 1 and 3, while the states 2 and 4 populate the second sublattice. In this checkerboard arrangement, each cation state in the ab-plane is surrounded by the perpendicular NN states. Note that there is no order in the system along the c-axis. The complete long-range order in the ab-planes is established in the orthorhombic phase. Here the ordering also occurs along the caxis, since the cations form stripes of the antiparallel (antiferroelectric) states (Figures 4 and S4). The obtained cation arrangements in all phases are in agreement with the reported structural data.43 This validates the proposed phase transition model and allows us to study the electric polarization of the cation subsystem. Figure 3b presents the temperature dependent electric polarization of MA+ cations in our model. The polarization averages out in the cubic phase due to the thermal fluctuations of the cation states. The situation is similar in the tetragonal phase, where the hopping of the cations occurs between the four remaining states. In the ordered orthorhombic phase, each ab-plane has nonzero polarization components P(ab) and P(ab) a b due to the checkerboard arrangement of the mutually perpendicular states. However, the net polarization of the whole crystal is zero as a result of the antiferroelectric-like arrangement of the cations along the c-axis (Figure S4). Thus, the calculated polarization vector components Pa, Pb, and Pc are zero for all phases indicating that the contribution of the MA+ cations to the low-temperature phases of MAPbI3 is of antipolar nature. We also noticed that if the ratio |ed/e1| ≳ 0.2, the lowtemperature phase of our model tends to split into several domains (see Figure S5). The energy of this multidomain order is slightly higher (more positive) than the energy of the ground state depicted in Figure 4. Such structures are characteristic to the phase transition models with substantial dipolar interactions.46−48 Note that a similar periodic domain arrangement of the ferroelastic origin was recently experimentally observed for MAPbI3 perovskite.27,28 To further support our MC simulation results, we performed the pyrocurrent measurements of the MAPbI3, MAPbCl3, and MAPbBr3 single crystal samples (see SI for sample preparation and experimental details). The obtained temperature dependence of the pyrocurrent is presented in Figure 5. For MAPbI3 compound, we observed an anomaly at 163 K, which corresponds to the tetragonal-orthorhombic phase transition. No anomaly was observed at the cubic-tetragonal transformation temperature. The pyrocurrent of MAPbCl3 exhibits two anomalies at 172 and 177 K which are related to the tetragonal-orthorhombic and cubic-tetragonal phase transitions. Two anomalies are also observed for MAPbBr3 at 146 and 153 K which are in agreement with the low-temperature phase transition temperatures. There could be several sources of the observed current such as occurrence of the electric polarization or formation of charged defects during the phase transitions. If we assume that the current has the former origin, we can use it to determine the temperature dependence of the electric

pi ̂ ·pĵ − 3(pi ̂ ·riĵ )(pĵ ·riĵ )

i,j

rij3

(1)

where p̂i denotes a unit vector of the electric dipole moment at the lattice site i. The total dipole moment of each state is p⃗i = p0p̂i, where p0 is equal for all states. The directions of the unit vectors p̂i coincide with the axes of the cubic lattice. For example, the S = 1 state at site i corresponds to p̂i = (0, 1, 0) unit vector (see Figure 2). ri⃗ j denotes the distance vector between two dipole moments p⃗i and p⃗j situated at the lattice sites i and j. The corresponding absolute value and unit vector of ri⃗ j are rij and r̂ij, respectively. The distance rij is calculated in units of the cubic lattice constant a. For the calculation efficiency, the sum in eq 1 is evaluated using a cutoff radius rc = 2a (see Figure 2c). The parameter ed describes the strength of the dipolar interaction. We also note that the dipole−dipole interactions affect all six cation states. The final Hamiltonian of our phase transition model contains the short-range strain-induced and long-range dipolar interactions: / = /s + /d

(2)

We solve this Hamiltonian using the MC simulations on a three-dimensional simple cubic lattice with the following set of interaction parameters: e1 = −43 meV, e2 = −13 meV, and ed = 8.6 meV (see SI for more details about the simulation procedure and choice of the parameters). First, we investigated the phase transition properties of our model by calculating the heat capacity CV. The temperature dependence of CV is presented in Figure 3a revealing two

Figure 3. Temperature dependence of (a) heat capacity and (b) electric polarization of our model obtained by MC simulations. The CV curves were calculated for ed = 0 (no dipolar interaction) and 8.6 meV. The polarization was obtained for ed = 8.6 meV. Other simulations parameters: e1 = −43 meV, e2 = −13 meV and L = 16.

anomalies at Tc1 = 315 K and Tc2 = 155 K. They correspond to the two structural transitions from the cubic to tetragonal and from tetragonal to orthorhombic phases. The phase transition at Tc1 is caused by the short-range interactions e1 and e2 (see Figure S2). The anomaly at Tc2 is absent for ed = 0 indicating that the dipolar interactions between the cations are responsible 4908


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Figure 4. Snapshots of MC simulations at different temperatures corresponding to the cubic (left), tetragonal (middle) and orthorhombic (right) phases. The MA+ cation states are color-coded, and the dipole moments are indicated by the arrows. The presented cubes were cut from a larger simulation (L = 20, see Figure S3).

Figure 5. Temperature dependence of the pyroelectric current and the corresponding electric polarization of (a) MAPbI3, (b) MAPbCl3 and (c) MAPbBr3 perovskites. The pyrocurrent of MAPbI3 was measured along the c-axis.

the whole phase transition sequence (summarized in Figure S6) and arrangement of the cations in all phases of this compound. This allowed us to investigate the temperature dependence of the electric polarization which revealed the nonferroelectric origin of the perovskite in agreement with many recent experimental studies. The ordering between the successive abplanes was found to be of the antipolar nature. We complemented the MC modeling by the pyrocurrent measurements of the MAPbX3 (X = I, Br and Cl) single crystal compounds. The obtained tiny saturation polarization for all halides supports our theoretical results. The presented model is based on the available structural data of the iodide compound, but we expect that it can be applied for the whole family of the methylammonium lead halides. Our model can be easily modified to study other important phenomena in these materials such as the recently proposed defect screening mechanism.18 This could help to further understand the relationship between the MA+ cations and high efficiency of the hybrid perovskite solar cells.

polarization (see Figure 5). In this way obtained saturation polarization is very small (≲ 0.2 μC/cm2) for all halides compared to the proper ferroelectrics such as inorganic BaTiO349 or hybrid [NH4][Zn(HCOO)3].50 Such a miniature polarization might arise from the ferroelastic domains that disturb the long-range order of the MA+ cations. The results of the pyrocurrent measurements are in agreement with our model and several experimental studies claiming that MAPbX3 perovskite family is antipolar (nonferroelectric).16−18,20,26,28 Finally, we address the ordering entropy of our model. The experimental data indicates that the total entropy change due to the structural phase transitions is ΔS = kB ln 24.21 The entropy change of our model is kB ln 6, because the ordering involves only six states of the MA+ cation. This discrepancy occurs, because our model does not take into account the rotation of the cation around the C−N axis, which requires additional model states. To maintain the cubic symmetry, each of the six cation states must have four substates that correspond to the 90° rotation around this axis (see Figure S6 for more details). This results in 6 × 4 = 24 states in the cubic phase as observed by the entropy measurements.21 The introduction of such additional degree of freedom in our model may change the phase transitions properties, but it would barely affect the arrangement of the dipolar moments and polarization of all phases. In conclusion, guided by the structural information, we constructed and solved the statistical model of the MA+ cation ordering in the MAPbI3 perovskite. Our model takes into account the short-range strain-mediated and long-range dipolar interactions between the cations. The model correctly describes

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b02239. Additional model, MC and sample preparation details (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: mantas.simenas@ff.vu.lt; Phone: +370 5 2234537. 4909


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Mantas Šimėnas: 0000-0002-2733-2270 Mirosław Mączka: 0000-0003-2978-1093 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Research Council of Lithuania (TAP LLT-4/2017). The authors thank A. Alkauskas for valuable discussion.

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