Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Saeed S. I. Almishal and
Ola Rashwan*

Penn State Harrisburg, USA. E-mail: ola.rashwan@psu.edu

Received
3rd October 2020
, Accepted 18th November 2020

First published on 17th December 2020

Inorganic metallic halide perovskites and cesium lead triiodide, CsPbI_{3}, in particular, have gained enormous attention recently due to their unique photovoltaic properties and low processing temperatures. However, their structural stability and phase transition still need an in-depth investigation to better optimize their optoelectronic properties. For sake of time and cost, Classical Molecular Dynamics (CMD) and first principles calculations are being used to predict the structure stability and phase transition of CsPbI_{3}. The major challenge of CMD is the choice of proper interatomic potential functions. In this paper, a new hybrid force field is being introduced, which integrates the embedded atomic potentials of Cs–Cs and Pb–Pb with Buckingham–Coulomb potentials. The Buckingham–Coulomb interatomic potential was solely employed as well. The outputs from both force fields were reported and compared to the experimental values. In fact, the new Hybrid Embedded Atomic Buckingham–Coulomb (EABC) potential reproduces, with a great degree of accuracy (within 2.5%), the structural properties, such as the radial distribution functions, interatomic separation distances, and the density. Also, it detects the phase transformation from an orthorhombic into a cubic crystal structure and the melting temperature at 594 K and 750 K respectively which agrees with the experimental values to within 1%. The new proposed hybrid potential proved to be accurate so it could potentially be used to infer the structure stability and the mechanical and thermal properties of the pure inorganic halide perovskites and the mixed halide perovskites as well which are used in various applications.

Halide perovskites have an ABX_{3} chemical formula, where A is a monovalent cation, B is a metal divalent cation and X is a halide anion. The archetypal perovskite of this family is methylammonium lead triiodide (MAPbI_{3}) which has been studied intensively. Solar cells based on MAPbX_{3} are highly attractive for their high absorption coefficient, tunable bandgap, low carrier recombination, and easy fabrication.^{5} The methylammonium (MA) cations cause complex dynamics because of their molecular orientation, molecular entropy and loss of stability.^{6} The choice of the A cations is important as it significantly influences the light absorption in the visible spectrum.^{7} Therefore, the MA cations could be substituted with larger cations, such as Cs^{+} or Rb^{+} for higher stability without sacrificing the optoelectronic properties as the lead halide sublattice is responsible for the direct bandgap.^{6,7}

Recently, cesium lead triiodide (CsPbI_{3}) has been getting attention along with the family of mixed halides of CsPb(I_{x}Br_{1−x})_{3}.^{8} CsPbI_{3} is an inorganic halide perovskite where a cesium cation (Cs^{+}) replaces the methylammonium cation (MA^{+}) of MAPbI_{3}. It has a more stable structure and unique photovoltaic properties. The black cubic polymorph of CsPbI_{3} (α-CsPbI_{3}) with Pmm structure and 1.73 eV bandgap, is the one used for the photovoltaic applications and was reported experimentally to be stable at a temperature higher than 593.15 K.^{9}

To better understand the effect of the temperature on the stability of the structure and the phase transition of inorganic halide perovskites over a relatively long-time, Classical Molecular Dynamics (CMD) is more favorable compared to the Density Functional Theory (DFT). Although there were numerous studies that used various first principles methods to understand the stability and electronic properties of halide perovskites,^{10–12} these methods are computationally expensive and unsuitable for large systems. Additionally, the stability of the structure over a relatively long time and various temperatures could not be effectively predicted by these methods. Classical Molecular Dynamics (CMD) offers a solution to model such large systems by treating the atomic movement classically through Newtons' laws.^{13–15} The availability and transferability of force fields, that describe the interatomic interactions among the compound constituents, pose a challenge in using CMD. Force fields are usually obtained by finding the proper fitting parameters that could reproduce the experimental structural properties, such as density and interatomic distances. These parameters are usually generated by first principle quantum mechanics calculations or empirical methods.^{13}

In classical molecular dynamics, the total energy E_{total} of the system is represented as the sum of all interactions between all the atoms in the system as shown in eqn (1)

(1) |

(2) |

The first hybrid potential to model MA^{+} based perovskites was developed by Mattoni et al.,^{16} ‘MYP’, by fitting on data from ab initio models. In MYP model, the inorganic–inorganic potential parameters were rescaled to Matusi MgSiO_{3} parameters.^{17} MA^{+} (CH_{3}NH_{3}^{+}) was represented by the standard GAFF force field that includes bonded interactions. (Pb–I) and (C–N) interactions were modelled by Buckingham–Coulomb-potential while (H–Pb) and (H–N) interactions were modelled by Coulomb–LJ potential. They also assumed that the interaction between C–Pb is the same as N–Pb and they both resemble Cs–Pb in CsPbI_{3}, and the same applies to C–I and N–I for Cs–I. The MYP fitting parameters were further enhanced by refining the MA^{+} atomic partial charges and the interactions of Pb–I to reproduce the density functional theory DFT calculations of the cohesive energy, Pb–I distortion energy, molecules reorientation energy and bulk modulus of a hydrostatic deformed cubic crystal. Further, the N–I interaction was refined to ensure the orthorhombic-to-tetragonal transition.^{16,18}

Mattoni et al.^{16} proved that the MYP calculations meet the experiments and DFT results on the important structural and vibrational properties of the material, such as energy as a function of lattice spacing.^{16} In studying the phase transition, the potential produced the general features of the transition from the orthorhombic crystal structure to the tetragonal structure and to the cubic crystal structure. Nevertheless, MYP was also used to determine the sources of phase transition in MAPbI_{3} which are mainly attributed to the dynamic behavior of organic cations in the lead iodide lattice.^{19} The MYP was further employed in studying the diffusion of point defects and it showed that the dominant mechanism of ionic transport in MAPbI_{3} is iodine diffusion.^{20}

For its acceptable accuracy, the MYP force field was used to study the behavior of different phases of MAPbI_{3} under various conditions. It was used to model the nucleation and interfacial mismatch during the vapor deposition of MAPbI_{3} on the TiO_{2} substrate.^{21} It was also used to study phonon scattering,^{22} carrier mobility,^{23} polaronic strain,^{24} and the qualitative characterization of the nature and the relaxation properties of the dielectric polarization.^{25} The use of MYP was not limited to 3D structures, it was also used to study the thermal conductivity of 2D MAPbI_{3} thin films.^{26} However, the resulted volume as a function of temperature curve from the MYP has discrepancies when compared to the experimental results because of the fact that the model ignored the charge redistribution and the covalent nature of lead iodide interactions.^{16}

Hata et al.^{6} studied MAPbBr_{3} using the same MYP potential developed by Mattoni et al.^{16} They used the same Pb–Pb, Pb–MA, and MA internal interactions as in MAPbI_{3}. They calibrated the newly introduced parameters based on five scaling factors for developing a force-field for MAPbBr_{3}. Three of which were for scaling; namely: α for coulomb–Buckingham parameters, β for the charges and γ for organic–inorganic interactions. Two additional factors resulted from the dynamics of phase transition which are A_{OI} and ρ_{OI} of Br^{−} (C, N). Although the developed model is simple, it reproduces, to a certain extent, the main static, and thermodynamic properties of MAPbBr_{3}.^{6,16} MAPbBr_{3} has almost the same molecular dynamic behavior as MAPbI_{3}. However, the volume of MAPbBr_{3} is consistently smaller than that of MAPbI_{3}. The change of volume–temperature curve slope for MAPbBr_{3} is at 140–170 K compared to 150–200 K for MAPbI_{3}. This is attributed to the orthorhombic-to-tetragonal phase transition as mentioned by Mattoni et al.^{16} Moreover, the anisotropy in the orthorhombic phase, longest to shortest lattice parameter ratio, of MAPbBr_{3} is slightly higher than that of MAPbI_{3} but still is underestimated relative to the experimental value. However, it is overestimated for MAPbI_{3}. In fact, for both MAPbBr_{3} and MAPbI_{3}, the calculated anisotropy does not change upon the phase transition from tetragonal to cubic which can be understood from the limitations of the two-body potentials for inorganic lattices. Still, by calibrating mixed I–Br interactions, it would be possible to model the large class of mixed systems MAPb(I_{x}Br_{1−x})_{3}.^{6}

Molecular dynamics simulations were used to study the effect of moisture on MAPI_{3} which is highly hydrophilic. A MYP-based force field (MYP1) for water–MAPI systems was developed to study the dissolution of MAPbI_{3} crystals in water.^{27} It was further used to clarify that the apparent hydrophobic experimental contact angle resulted from the surface degradation during measurement.^{28} Despite the accuracy of the models developed by Mattoni et al.^{16} and Hata et al.,^{6} these models cannot predict the complex crystal structure transitions and they are not transferrable to mixed halides perovskites MAPb(I_{1−x}Br_{x})_{3}.

Handley and Freeman^{29} also presented a set of interatomic potentials for modelling MAPbI_{3} that included short-range interactions for the lead and iodide interactions to improve the degree of transferability. Artificial intelligence algorithms, such as artificial neural networks (ANNs) and genetic algorithms (GAs) were employed to develop force fields that represented the mixed halide perovskites. Chen and Pao^{30} used artificial neural networks (ANN) to model complex structures and transitions. They demonstrated that by developing an ANN potential for MAPbI_{3} as an example. Marchenko et al.^{31} developed a semi-empirical model that represents the mixed halides hybrid perovskites and they used it to calculate Gibbs free energy and thermodynamic properties. Balestra et al.^{32} used genetic algorithms to adjust the Coulomb–Buckingham pair atomic potentials to model the mixed halide systems of CsPb(I_{x}Br_{1−x})_{3}. They used their modified potential values to reproduce the DFT calculations of the lattice geometric distortions energies. However, their modified force field was fitted only to cubic structures of CsPbI_{3}, CsPbBr_{3,} and mixed CsPbI_{3}Br with the aim to be transferable between them. The radial distribution function of the cubic CsPbI_{3} was successfully obtained and matched the experimental results.^{32} However, the new derived force field acceptably agreed with experiments and previous theoretical studies of structural and dynamic properties, particularly, for the pure structures, i.e., the non-mixed, in their cubic crystalline structure form.

Although, the inorganic metal halide perovskites, particularly CsPbI_{3}, have unique properties and better stability than MAPbI_{3}, their structural properties, and phase transitions are not well-characterized and further investigation is needed. Therefore, the objective of this work is to apply molecular dynamics to investigate the structural properties and phase transformation of CsPbI_{3} by applying the proper force field. We achieved this by introducing a novel hybrid force field which is referred to as “Hybrid EABC”, where EABC stands for Embedded-Atomic-Method and Buckingham–Coulomb potential. We compared the outcomes of using both Coulomb–Buckingham potential and Hybrid EABC in reproducing the experimentally reported and the ab initio structure's radial distribution functions and phase transitions of CsPbI_{3} with an ultimate aim to gain in-depth understanding of the CsPbI_{3} to further enhance its properties.

Fig. 1 The resulting orthorhombic structure of CsPbI_{3} from DFT. The structure is composed of 4 Cs atoms (in red color), 4 Pb atoms (in blue color) and 12 I atoms (in yellow color). |

Buckingham–Coulomb potential (U_{BC}) is numerically more stable than the Lennard-Jones potential.^{36} In Buckingham–Coulomb potential (U_{BC}), the long-term coulomb interaction is added to Buckingham potential as shown in eqn (3).

(3) |

The potential parameters for the six interactions are required to completely define the force field between the constituent ions of CsPbI_{3} perovskite. For the parameters of Cs–Cs and Cs–I Interactions, the Buckingham parameters were obtained from Sangster and Atwood.^{38} For Cs–Pb, interaction, we used Mirskaya technique which is based on the geometric mean combining rule for the total interaction energy of two non-bonded atoms for the Buckingham potential and shown in eqn (4) and (5).^{39}

(4) |

(5) |

For C_{Cs–Pb}, an iterative algorithm was developed to find the ideal value. This value was approximately the same value for C_{Cs–Pb} that was obtained by Mirskaya^{39} method from C_{Cs–Cs} and C_{Pb–Pb} by Balestra et al.^{32} For the parameters of Pb–Pb, Pb–I, and I–I interactions, they were set to be equal to the ones in MYP.^{16,18} The effective charge of Cs^{+} was assumed to be the same charge of MA^{+} as in MAPbI_{3}.^{18} These interactions along with the effective charges are summarized in Table 1.

Interaction | A (eV) | ρ (Å) | C (eV Å^{6}) |
Ion | Effective charge |
---|---|---|---|---|---|

Cs–Cs | 3.584 × 10^{16} |
0.0843 | 240.34 | Cs | 1.36 |

Cs–Pb | 3.3068 × 10^{11} |
0.10519 | 480.0 | ||

Cs–I | 4913.000 | 0.3814 | 480.06 | Pb | 2.03 |

Pb–Pb | 3051120.5 | 0.131258 | 0.0 | ||

Pb–I | 4488.05 | 0.321737 | 0.0 | I | −1.13 |

I–I | 988.42 | 0.482217 | 30.22285 |

Therefore, in this work, we developed a new potential of CsPbI_{3} by superimposing the Embedded Atomic Method (EAM) to BC pair potential to model both Cs–Cs and Pb–Pb interactions to reflect the metallic nature of these bonds. The EAM potential was first developed in 1984 by Daw and Baskes to study the hydrogen embrittlement in metals.^{41} EAM potential (U_{EAM}) adds the embedding energy, which simply represents the energy required to embed positively charged atomic cores into the electron density, to the pair atomic potential. The EAM is presented schematically in Fig. 3.

The EAM potential can successfully model the cohesion, defects, surface relaxation, and ground-state impurity energies resulting in overcoming the limitations of pair potentials.^{13,40,42} In general, it improves the potential performance dramatically and still could be applied to large systems. The EAM potential (U_{EAM}) is estimated using eqn (6) and (7) in terms of the pair potential and embedding energy as a function of electron density.

(6) |

(7) |

In our new proposed Hybrid EABC potential, for the interactions between Cs–Cs and Pb–Pb, the Buckingham part of the potential shown in eqn (3) is replaced by the EAM potential shown in eqn (6)^{43,44} while the Coulomb potential is still imposed. The new Hybrid EABC potential parameters are summarized in Table 2.

The CMD simulations outputs were the radial pair distribution function, g_{αβ}(r), and the change of density with the temperature. The g_{αβ}(r) was calculated after the relaxation at room temperature conditions using eqn (8).

(8) |

The radial distribution function is a structural descriptor that represents the probability distribution of β particles surrounding another specific α particle. The radial distribution function exhibits a series of discrete peaks in the case of a single crystal. In general, the magnitude and the location of the peaks depend on the exact crystal structure.^{49}

The radial distribution functions of bulk CsPbI_{3} at 300 K using the Buckingham–Coulomb potential and Hybrid EABC potential were obtained using 100 bins and illustrated in Fig. 5 and 6 respectively. One significant difference between the outcomes of using the two potentials occurs in the Pb–I interaction, where the Buckingham–Coulomb peak is shifted to the right with ∼0.15 Å compared to the Hybrid EABC peak. Moreover, both peaks of Cs–Pb and Pb–I from the Hybrid EABC potential agree with the radial distribution functions resulted from the DFT calculations shown in Fig. 7, which implies that the inclusion of EAM potential in the Hybrid EABC helps to maintain the integrity of the initial orthorhombic structure. Moreover, the peaks resulting from using the hybrid potential are closer to the peaks of the radial distributions functions which were obtained experimentally^{50} and are shown by the dotted red lines in Fig. 5 and 6.

Fig. 5 The radial distribution function of the equilibrated structure with Buckingham–Coulomb at 300 K and g(r) with 100 bins (yellow: I, red: Cs and blue: Pb). The experimental values are obtained from ref. 50. |

Fig. 6 The Radial distribution function of the equilibrated structure with Hybrid EABC at 300 K and g(r) with 100 bins (yellow: I, red: Cs and blue: Pb). The experimental values are obtained from ref. 50. |

Fig. 7 Radial distribution function of the structure that resulted from DFT and inputed to MD. This RDF was generated with 1000 bins. |

The interatomic separation distances between the different ions using the Buckingham–Coulomb and Hybrid EABC potential functions were obtained from RDF with 1000 bins for higher accuracy and are reported in Table 3. When comparing the outcomes of the two force fields, the largest deviation occurs at the Cs–Pb interaction, where the Buckingham–Coulomb Cs–Pb peak is shifted ∼0.4 Å to the left of the Hybrid EABC peak. Whereas the least deviations occur at Cs–Cs and I–I peaks with a deviation of ∼0.01 Å.

Interaction | Interatomic distance (Å) | |
---|---|---|

Coul–Buck | Hybrid EABC | |

Cs–Cs | 4.4820 | 4.4700 |

Cs–Pb | 5.3820 | 4.9860 |

Cs–I | 3.6900 | 3.7500 |

Pb–Pb | 4.4820 | 4.5180 |

Pb–I | 2.9100 | 3.0540 |

I–I | 4.5180 | 4.5060 |

In fact, the new proposed Hybrid EABC results in Pb–I bond lengths of ∼3.05 Å which agrees with experimental findings reported by Straus et al.^{50} with an error of less than 1%. Also, it proves that the change in lattice parameters with temperature must originate from tilting in the octahedra of Pb–I. The accuracy of modelling Pb–I interaction is crucial in modelling lead iodide perovskites as the electronic and optical properties are greatly dependent on this interaction.^{7}

It is worth noting that when the structure was equilibrated with the Hybrid-EABC force field but with omitting the Cs–Cs and Pb–Pb coulomb interactions, the orthorhombic structure clustered into CsI and PbI_{2} as shown in Fig. 8. This demonstrates the experimental method of mixing CsI and PbI_{2} (ref. 9) to produce CsPbI_{3} but in reverse. Also, this implies the significance of coulombic interactions between Cs–Cs and Pb–Pb in stabilizing the structure.

Fig. 8 Clustering of CsPbI_{3} as CsI and PbI_{2} when the coulombic interactions are neglected between cesium ions and between lead atoms. |

The scatter plots of the X-positions vs. Z-positions of CsPbI_{3} ions within the supercell at 0 K, 300 K and 600 K are shown in Fig. 9. A clear rattling behavior appeared in the scatter plot of the X-positions vs. Z-positions of at 300 K when compared to 0 K. This behavior can be attributed to the distortion of Cs atoms and Pb–I octahedra. The highly distorted and irregular behavior of Cs ions can also be observed in our Cs–Cs interaction in the RDF results shown in Fig. 5 and 6. This agrees with the findings in ref. 50 that Cs cations become distorted, occupy two positions and rattle within the structure causing the instability of CsPbI_{3} at ambient conditions. At 600 K, the increase in volume which is observed from the increase in the scattered positions along the X-axis indicates the transition into a cubic crystal structure and it would be discussed in more detail in the phase transformation section.

Fig. 9 Scatter plot of X positions vs. Z positions for the 1920 atoms at (a) 0 K, (b) 300 K and (c) 600 K at one time for the Hybrid EABC model. |

When the new hybrid EABC was employed, the density vs. temperature curve exhibits an abrupt change of density at different temperatures indicating phase transformation as shown in Fig. 11. At 300 K, CsPbI_{3} is stable as an orthorhombic structure with a density equal to 5.2558 g cm^{−3}. Then, the density decreases gradually as the temperature increases to ∼560 K and an abrupt decrease in density starts at ∼610 K and it continues to decrease till it reaches 4.81 g cm^{−3} at 630 K. This could be attributed to the phase transformation from the orthorhombic to cubic structure, i.e. α-CsPbI_{3}. This agrees within 1.5% with the experimental result by Trots et al.^{51} which stated that CsPbI_{3} undergoes orthorhombic-to-cubic phase transition through two stages. The first starts at 563 K where orthorhombic and cubic structures coexist and the other is at 602 K where only cubic structure is present. During this phase transition, an increase in volume by 7.5% occurs which is very comparable to the experimental value of 6.9% reported in ref. 51.

Fig. 11 The density of CsPbI_{3} as a function of temperature calculated by Hybrid EABC. The scatter points represent measured points from the CMD model and the orange curve is obtained by regression fitting. The red dashed lines represent iso-temperature lines obtained from experimental values in ref. 9,51,52. |

At 630 K, the density drops from 5.1 to 4.81 g cm^{−3} while the system's Nose–Hoover thermostat was trying to maintain the system temperature at 600 K. This decrease in density along with the increase in volume was also observed in the scatter plot of the X-positions vs. Z-positions at 600 K shown previously in Fig. 9. Therefore, the system would eventually equilibrate at a density of 4.81 g cm^{−3} at 600 K. The resulting structure is stable from ∼593 K. The average temperature of the system from the beginning of the transformation till the end of the equilibration at the new phase is 594.6 K. These findings also perfectly match the experimental densities and the temperature reported by Wang et al.^{9} and Trots et al.^{51} and indicated by the dashed lines in Fig. 11. Further heating of the system results in a second change of slope at ∼750 K which might be attributed to melting as reported by Sharma et al.^{52}

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