Theoretical Study of Stability of Halogen‐Defective Trihalide Monolayers: Cases of AlI3, AsI3, and IrBr3

A theoretical study is conducted with three MX 3$_{3}$ monolayer 2D materials (AlI 3$_{3}$ , AsI 3$_{3}$ , and IrBr 3$_{3}$ ) on their electronic properties and how a halogen monovacancy affects their thermostability. Density functional theory (DFT) calculations are run to obtain the band structures and phonon dispersions for both pristine and defective structures. It is shown that AlI 3$_{3}$ and AsI 3$_{3}$ have indirect bandgaps of 2.40$2.40$ and 2.23$2.23$  eV, respectively. IrBr 3$_{3}$ has a direct bandgap of 1.65$1.65$  eV. Phonon dispersions indicate that they are all thermodynamically stable in pristine state, but their defective counterparts are not. Ab‐initio molecular dynamics (AIMD) simulations are conducted for defective ones to further investigate their stability. It is found that AlI3 and AsI3 layers are decomposed while IrBr3 layer is bent. Further investigations are conducted by analyzing the bond energies and bond lengths of the three materials. It shows that AlI 3$_{3}$ and AsI 3$_{3}$ have lower bond energy and longer bond length, which makes them dissociated at the ambient temperature while higher bond energy and shorter bond length keep IrBr 3$_{3}$ stabilized and enable its displacive phase transition in displacive limit.

DOI: 10.1002/pssb.202300001 A theoretical study is conducted with three MX 3 monolayer 2D materials (AlI 3 , AsI 3 , and IrBr 3 ) on their electronic properties and how a halogen monovacancy affects their thermostability. Density functional theory (DFT) calculations are run to obtain the band structures and phonon dispersions for both pristine and defective structures. It is shown that AlI 3 and AsI 3 have indirect bandgaps of 2.40 and 2.23 eV, respectively. IrBr 3 has a direct bandgap of 1.65 eV. Phonon dispersions indicate that they are all thermodynamically stable in pristine state, but their defective counterparts are not. Ab-initio molecular dynamics (AIMD) simulations are conducted for defective ones to further investigate their stability. It is found that AlI 3 and AsI 3 layers are decomposed while IrBr 3 layer is bent. Further investigations are conducted by analyzing the bond energies and bond lengths of the three materials. It shows that AlI 3 and AsI 3 have lower bond energy and longer bond length, which makes them dissociated at the ambient temperature while higher bond energy and shorter bond length keep IrBr 3 stabilized and enable its displacive phase transition in displacive limit.
not applied on defective ones, partly because of phonon anharmonicity which cannot be captured by this method. [20] In addition, the phonon dispersion obtained from DFT study cannot provide the structural variation upon temperature changes. Although there are already tools and software [21][22][23][24] offering more comprehensive approaches that relate AIMD and phonon dispersion at different temperatures, a simple AIMD simulation is always a good choice to complement it, where higher-order force constants and larger atomic displacements are considered. [20,21] In this study, we focus on three 2D MX 3 materials, AlI 3 , AsI 3 ð001Þ, and IrBr 3 ð001Þ, by studying the electronic structures and stability of their pristine monolayers and defective ones with a halogen vacancy. The band structures and the phonon dispersion of the pristine structures were calculated. While they were also computed for the defective structures, additional AIMD simulations were conducted as well to validate the results from phonon dispersion. The difference between structural deformation noticed in MD was explained by the calculation of bond energy and bond length, which affect the solid materials' melting point, while the bending behavior of IrBr 3 was explained according to displacive phase transition theory.

Results and Discussion
The structure of monolayer AlI 3 is illustrated in Figure 1a. Trihalides MX 3 have a structure belonging to space group P-31m.
A monolayer contains three layers of atoms stacked in an ABC or ABA order, in which metallic atoms are located in the middle. Each nonhalogen atom is surrounded by six halogen atoms that form an edge-sharing octahedron. The fully relaxed lattice parameters are listed in Table 1. Compared to the bulk-state structures from Materials Project, [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] a and b of the monolayers are 4% and 1% smaller than the ones of the bulk state for AsI 3 and IrBr 3 respectively. The phonon dispersions and band structures of the pristine AlI 3 , AsI 3 , and IrBr 3 monolayers are shown in Figure 1c-h. There are no negative modes in the phonon dispersion of AlI 3 and IrBr 3 , which implies that they are thermodynamically stable, in line with the collected data in C2DB database. [41] However, small negative modes appear around the Γ point for AsI 3 . Since the structure was thoroughly relaxed, its appearance is due to insufficient cutoff energy. Nevertheless, Figure 1. a) The structure of AlI 3 (the gray atoms are aluminum, and the purple ones are iodine) and its primitive cell delimited by the black borders. b) The 2S reciprocal lattice of M 2 X 6 monolayer structures, in which black arrows indicate the basis vector and the red rectangle indicates the k path selected for the calculations of phonon dispersion and band structure. c-e) The phonon dispersion of AlI 3 , AsI 3 , and IrBr 3 respectively. f-h) The band structures and DOS of AlI 3 , AsI 3 , and IrBr 3 respectively, in which the dashed black lines are where Fermi energy lies. www.advancedsciencenews.com www.pss-b.com these modes can be eliminated by adjusting the parameters in the calculation. Figure 1 and the data in Table 1 show that they have a bandgap of around 2 eV, which is suitable for absorbing the visible light whose spectrum spans from 1.59 to 3.26 eV. The bandgap values of AsI 3 and IrBr 3 are close to those of their bulk-state counterparts, which are 2.24 and 1.56 eV. For all the samples studied here, the CBM and VBM embrace the oxidation potential of H 2 O/O 2 and the reduction potential of CO 2 /CH 4 , which are 0.82 V versus normal hydrogen electrode (NHE) and À0.24 V versus NHE, implying that they are all suitable candidates for the application of CO 2 reduction regarding the electronic structure. The density of states (DOS) projected on elements are the lines with shadows in the DOS plots, and the DOS of each orbital is plotted in Figure 2. Below the Fermi level, the valence band close to VBM of AlI 3 is dominated by the iodine 5p states, the same as in the case of AsI 3 . The states in conduction band near CBM are dominated by Al 3s states for AlI 3 and As 4p states for AsI 3 . For IrBr 3 , while Ir 5d and Br 4p are the main orbitals near CBM and VBM, the contribution from Ir 5d always is the majority. In addition, the large overlap between these two orbitals indicates that the interatomic interactions between those atoms are strong. The bandgap of AsI 3 perfectly agrees with the value in the works of Lai et al. [6] and Liu et al. [12] . Compared to the values obtained with PBE functionals in C2DB, [41] 2.09, 1.91, and 1.57 eV for AlI 3 , AsI 3 , and IrBr 3 , respectively, they are 5-16% larger. Various factors may contribute to it, considering that the code, as well as the pseudopotential files used, are not identical. The CBM of AlI 3 and AsI 3 are at Γ point but their VBMs are not; hence, they have indirect bandgaps. The Brillouin zone and the k-path are illustrated in Figure 1b. The VBM of AlI 3 is located at reciprocal coordinate ð0.098, 0.098, 0Þ Å À1 , on the path from K to Γ, while that of AsI 3 is at ð0.207, 0, 0Þ Å À1 , on the path from Γ to M. On the contrary, IrBr 3 has a direct bandgap with CBM and VBM at ð0.495, 0, 0Þ Å À1 , a position close to M point, unlike its bulk-state counterpart which has an indirect bandgap. It is well known that the use of generalized gradient approximation in DFT usually underestimates the bandgaps because of the self-interaction error, [42] and for the same reason, the absolute valence band position is also affected. [43] Therefore, the results displayed here may be somewhat different from the ground truth, while still being valuable.
To study the effect of monovacancy on the structures of these three materials, an iodine vacancy or a bromium vacancy was introduced by removing a corresponding atom on the 4 Â 4 supercell. The formation energy was calculated with the following formula where E defect is the total energy of the defective structure, E X2 is the energy of an isolated halogen molecule in vacuum, and E pristine is the total energy of the corresponding 4 Â 4 supercell. The values for AlI 3 , AsI 3 , and IrBr 3 are 2.20, 1.50, and 2.63 eV, respectively. Compared to 7.02 eV, [44] the formation energy of a sulfur vacancy on MoS 2 monolayer, those defects are easier to form. The band structure and the DOS for defective structures are shown in Figure 3. Though DOS analysis, gap states are present after removal of a halogen ion in the defective structure of each material. Four gap states are observable in the bandgap of AlI 3 : two at 0.11 eV with opposite spins, 2.02 eV with spin-up, and 2.27 eV with spin-down respectively. Two gap states of AsI 3 are at 1.0 and 1.69 eV with opposite spin respectively. For IrBr 3 , three gap states are respectively at 0.33, 1.08 eV with spin-up and 0.80, 1.45 eV with spin-down. These states can significantly affect the optical and transport properties of the materials. Phonon dispersions were calculated after the structures were optimized, which are shown in Figure 3. In the phonon dispersion of AlI 3 , large negative frequency values are seen spanning the whole q-path. Beside the lowest one, there is more than one band that sinks between the two Γ points at the ends of the qpath. Small negative frequencies appear in the phonon dispersion of AsI 3 and IrBr 3 as well, although the difference is that the band below zeros for AsI 3 is a convex function along the q-path while that for IrBr 3 is a concave function.
Phonon dispersion offers an apparently straight and convenient way to reveal the dynamic stability of the materials, but www.advancedsciencenews.com www.pss-b.com it does not tell how the structure will deform and the degree of deformation. By removing a halogen atom on the monolayer, we create a vacancy, as well as change the covalent bonds to the neighboring atoms. Taking AlI 3 as an example, an aluminum atom is bonded with the six neighboring iodine atoms by sharing its valence electrons in 3s 2 3p 1 orbitals. Three valence electrons in total are split into six bonds, 1=2 for each. But when the vacancy is created, 1=2 valence electron from the broken bonds is redistributed to the remaining five bonds, since there is no significant change in the charge of the aluminum atoms. The situation is similar for AsI 3 and IrBr 3 as well. The only difference is that As and Ir have 5 electrons in 4s 2 4p 3 and 9 electrons in 6s 2 5d 7 respectively. The creation of a vacancy breaks a connection between the nearby nonhalogen atoms, which makes the constraints on the two sides of the monolayer different. Although the optimized structures from DFT remain almost unchanged from the original pristine ones, which are dynamically unstable as indicated by phonon dispersion, the phase transition process is not clear. Whether the structures will be dissociated or evolve into another thermodynamically stable phase remains unknown. Thus, other methods are needed to reveal the dynamic stability of the defective structures such as AIMD.
The trajectories of temperature and total energy and the final structures of the AIMD simulations are presented in Figure 4. The fluctuation of energy decreases over time, as well as the temperature evolution for all three materials, indicating that equilibrium is reached within the time durations of the simulations.
There is an increase in energy at the initial stages of the energy curve, which implies that the relaxation process for these structures is endothermic at ambient temperature. The structure of AlI 3 is broken after a 5 ps run at ambient temperature. The monolayer started to deform and to decompose into AlI 3 molecules. The situation is even worse for AsI 3 , where multiple separated AsI 3 molecules are visible and the whole structure has already broken down.
On the contrary, IrBr 3 monolayer behaves dissimilarly since it surprisingly was not dissociated and was only bent. The layer is deformed into an arc around x-axis, with a radius in the range of 70-75 Å. Although this value is much larger than that of a typical nanotube, whose radius is in the magnitude of Angstrom, this change may still bring some novel features to this 2D material.
To find out why the deformation of the three materials behaves differently, the bond energies were calculated. It can be calculated by dividing the energy of formation E f by 2, because the creation of a halogen vacancy is done by removing a halogen atom and breaking two bonds that connect it with the neighboring two nonhalogen atoms. [45] It gives us the values listed in Table 2, along with the bond lengths d. The bond energy of Ir─Br is 50% larger than that of Al─I and As─I. From the perspective of thermal stability, the melting temperature of the material that has higher bond energy is higher than that of the material which has lower bond energy. This relation is illustrated in ref. [46]. In our case, it can be easily seen that IrBr 3 keeps its cohesion, while AlI 3 breaks. In Figure 4g, one AlI 3 www.advancedsciencenews.com www.pss-b.com molecule leaves but the overall monolayer shape is still conserved. The structure of AsI 3 is utterly destroyed, with at least two AsI 3 molecules clearly dissociated in Figure 4h. The snapshots of these two materials show different degrees of dissociation. The isolated state of the monolayers may explain this observed instability and bilayer (defective/pristine) AIMD simulation was performed to try and conclude as to the stability of the defective monoalayer with support from another. The simulations were run with the same parameter setup for a shorter time, but the results remain the same as the interactions between layers are negligible (%0.002-0.003 eV atom À1 ). Their energy and temperature profiles are shown in Figure S1, Supporting Information. The closer a material gets to its melting point, the more it starts to degrade before losing its solid structure entirely. From this, associated with the order of the bond energies, we can propose a relative order of melting points of the three materials with respect to 300 K It is in agreement with the bond length order dðAs À IÞ > dðAl À IÞ > dðIr À BrÞ which is a strong indicator of the strength of interaction. The slight discrepancy in the bond energy order between As--I and Al--I is likely to come from a different type of bonding, Al--I being the mostly ionic (ΔEN ¼ 3.546 eV) and As--I being the mostly covalent (ΔEN ¼ 1.459 eV). [47] Overall, the temperatures for which the decomposition of AlI 3 and AsI 3 starts are lower than that of IrBr 3 . This is the reason why IrBr 3 is not dissociated in the AIMD run. The critical temperatures for dissociation T c of pristine monolayers were calculated where the difference of their Gibbs free energy from the ones of their corresponding MX 3 molecules is ΔG ¼ 0. They are estimated to be 1500, 500, and ≫ 2000 K for AlI 3 , AsI 3 , and IrBr 3 respectively. We obtained T c ðAsI 3 Þ < T c ðAlI 3 Þ ≫ T c ðIrBr 3 Þ, in agreement with the order proposed above. The changes of ΔG versus temperature are plotted in Figure S2, Supporting Information. The trend is roughly consistent with the results we got from the AIMD simulation of the three materials, although the exact values of T c are much higher than 300 K, losing its meaning in this study. This discrepancy of values is understandable because a lot of factors have been taken into consideration in AIMD, for example, the fluctuation of temperature and the vibration of atoms, while the calculation of ΔG was not (obtained at 0 K). Another interesting point that should be noticed is the rise in energy at the initial stages of the AIMD simulation, although IrBr 3 was not dissociated. This implies that the bending behavior of the IrBr 3 monolayer is an Figure 4. The trajectories of energy and temperature of AIMD for monovacancy monolayer a,b) AlI 3 , c,d) AsI 3 , and e,f ) IrBr 3 respectively. g-i) The snapshots of the structure at the end of each AIMD for AlI 3 , AsI 3 , and IrBr 3 , respectively. www.advancedsciencenews.com www.pss-b.com activated process. The AIMD trajectory of the IrBr 3 shows that the monolayer was bent upward and downward over time (see animation in Suppoting Information), which is a typical character of displacive phase transition [48] when the temperature is higher than the transition temperature T c . It indicates that there are two local potential minimums along the axis out of the plane. Although the energy barrier V 0 between them is not calculated, it can be estimated to be less than 0.026 eV (T % 300 K), two magnitudes lower than the Ir─Br bond energy, meaning that this phase transition is within the displacive limit (s ¼ jV 0 j=k B T c ( 1). In this domain, the strength of the interactions between neighboring atoms is much stronger than that of the potential wells around the local minimums.

Conclusion
In summary, we theoretically studied the band structures and phonon dispersions of three MX 3 materials, AlI 3 , AsI 3 , and IrBr 3 . The effect of halogen monovacancy on their electronic properties and thermostability. The pristine monolayers are all stable and in agreement with the data in C2DB. They have a bandgap of 2.40, 2.23, and 1.65 eV, which are photocatalytic candidates for CO 2 /CH 4 reduction under visible light. The creation of a monohalogen vacancy introduces an intermediate energy level and magnetic properties that can have potential use. Although the phonon dispersion with the structures optimized by DFT shows that all the monolayers are unstable with the vacancy, the ambient-temperature AIMD results show that the halogen vacancy unstabilizes AlI 3 and AsI 3 but bends the IrBr 3 layer. Further study with bond energy and bond length reveals that the bond strength plays a key role in keeping IrBr 3 stabilized.
The oscillating behavior along the out-of-plane axis of IrBr 3 is discovered and explained by a displacive phase transition in the displacive limit. It has been studied that the introduction of vacancies may wrinkle or fold a 2D material similar to the situation here. MoS 2 monolayer, for example, by controlling the concentration of sulfur vacancies on one side, the layer can wrinkle or even fold. [49] By doing this, the molybdenum atoms in the middle of the monolayer are exposed for adsorption. Not only can it lower the adsorption energy of molecules, but it can also promote its reactivity. This property can be further improved by the addition of metal atoms on the surface, which interacts with the charge transfer at the vacancy sites. [50,51] . In addition, the desulfurization technique is also used in the study of self-assembly materials. [52] . Hence, further study may be carried out on the bent IrBr 3 for its potential physical and chemical properties induced by this structural deformation.

Computational Methodology
The crystal structure files of AlI 3 , AsI 3 , and IrBr 3 were adopted from C2DB database [41] for further optimization. First-principle DFT calculations were conducted using Vienna ab-initio Simulation Package (VASP) [53][54][55] with PBE functionals. [56,57] The cutoff energies for all calculations were set to 500 eV. A vacuum slab of 20 Å was used. Lattice parameters were optimized for all pristine monolayer structures with fixed cell volume and 8 Â 8 k-point grid. The convergence criterion of the electronic step was set to 10 À6 eV, while that of the ionic step was set to 0.02 eV Å À1 .
Then a self-consistent field (SCF) calculation was conducted for each monolayer structure with an increased k-point grid of 10 Â 10. It was followed by the calculations of band structure and DOS. The electronic convergence was set to 10 À8 eV.
All three defective monolayer structures were obtained by removing an iodine or a bromine atom from the 4 Â 4 supercells. The convergence criterion of the electronic step in geometry optimization was met when the energy difference between two continuous steps was less than 10 À8 eV, while that of the ionic step was met when the force on each atom was less than 0.001 eV Å À1 . The subsequent SCF, band structure, and DOS calculations were done with electronic convergence of 10 À6 eV. Only Γ point was considered. The POSCARs of the defective monolayers were uploaded as POSCAR_d-AlI3, POSCAR_d-AsI3, and POSCAR_d-IrBr3 in the Supporting Information.
Phonon dispersions of pristine structures were computed through the finite difference method using 4 Â 4 supercells of the primitive cells in a single calculation while taking into consideration the symmetry. The phonon dispersions of the defective structures were obtained with the help of Phonopy. [58] The convergence criteria of the electronic and ionic step were 10 À6 and 10 À5 eV respectively.
AIMD simulations with defective monolayers were conducted with VASP employing its on-the-fly machine learning force field. The time step was 2 ps and each simulation was run for 5 fs at 300 K. NVT ensemble was used with Nose-Hoover thermostat. Only Γ point was considered.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.