Theoretical Investigations into the Different Properties of Al-Based Fluoroperovskite AlMF3 (M = Cr, B) Compounds by the TB-MBJ Potential Method

Al-based fluoroperovskites compounds AlMF3 (M = Cr, B) are investigated computationally and calculated their elastic, structural, optical, and electrical properties in this study utilising TB-MBJ potential (also GGA+U for AlCrF3) approximations, according to the Birch Murnaghan Equation curve and tolerance factor, these material are structurally cubic and stable. The IRelast algorithm is used to forecast elastic properties, and the outputs show that these compound are mechanically stable, anisotropic and ductile. AlBF3 has a metallic nature and overlapping states, while AlCrF3 have a narrow indirect band gap at (X-M) points of symmetry, with band gaps of 0.71 eV for AlCrF3 and zero eV for AlBF3. The partial and total density of states are being used to determine the influences of different basic states to the conduction and valence bands (TDOS & PDOS). Investigation of Optical properties shows that these compounds have low refractive index and high absorption coefficient, conductivity, reflective coefficient at high energy ranges. Owing to the indirect band gap, the applications of these compounds are deemed in conducting industries. Here we are using these compounds for first time and are examined using the computational method, which delivers a complete view into the different properties.


Introduction
Fluoroperovskites compounds are familiar and having the natural chemical formula ABF 3 , where cations are "A" and "B" with positive and Fluorine element having negative charge is an anion. Fluoroperovskites are a fascinating group of materials with a stable mechanically crystal structure and good optoelectronic characteristics ranging from semiconducting material having energy ranging from 1-4 eV and insulating materials with energy (above 4 eV) in nature. Fluoroperovskites have received a lot of interest recently due to its usage in a variety of industries, including semiconductor production,  The RMT values for Cr, B, and Al were 1.74, 1.74, and 2.5, respectively, while the values of RMT for F in the AlCrF3 and AlBF3 were 1.77 and 1.74, respectively. In the Muffin tin spheres, the wave function is stretched in harmonics up to lmax= 10 because there are 2000 K-points and 12 Gmax. The valence bands and core had a quite large 6.0 Ry energy difference.

Electronic Properties
In this unit, the energy bands structures and DOS of compounds under investigation are reported. In the primary Brillion area of AlMF3 (M=Cr, B), the estimated bands struc ture and symmetry directions are shown in Figure 3. Since no zero energy should exist a or close to the maxima location of the valence band. Using the said potential approach the valence band maxima of the compound AlBF3 overlap the minima of conduction band and zero band gap while the band gaps for the compounds AlCrF3 were determined to be 0.71 eV, respectively. The valence band maximum in both cases crosses the fermi energy level, indicating that both compounds have metallic properties. The indirect band nature of AlCrF3 is shown by the electronic band structure in Figure 3a, which shows that the conduction group of states minima of AlCrF3 occur at the equilibrium point X and the valence band peak occur at the point M (a).The metallic character of the AlCrF3 combina tion is obvious. Simultaneously the GGA + U were applied for the AlCrF3, due to the tran sition element. The effective Hubbard parameter (Ueff) in the AlCrF3 system was adjusted to 4 eV for the Cr-3d orbital. That shows that the nature of the material changed from metallic to half metallic nature (see Figure 3b). The minima of conduction band of AlBF and the maxima of valence band occur at R of Brillion Zone, which demonstrates the direc  The RMT values for Cr, B, and Al were 1.74, 1.74, and 2.5, respectively, while the values of RMT for F in the AlCrF3 and AlBF3 were 1.77 and 1.74, respectively. In the Muffin tin spheres, the wave function is stretched in harmonics up to lmax= 10 because there are 2000 K-points and 12 Gmax. The valence bands and core had a quite large 6.0 Ry energy difference.

Electronic Properties
In this unit, the energy bands structures and DOS of compounds under investigation are reported. In the primary Brillion area of AlMF3 (M=Cr, B), the estimated bands struc ture and symmetry directions are shown in Figure 3. Since no zero energy should exist a or close to the maxima location of the valence band. Using the said potential approach the valence band maxima of the compound AlBF3 overlap the minima of conduction band and zero band gap while the band gaps for the compounds AlCrF3 were determined to be 0.71 eV, respectively. The valence band maximum in both cases crosses the fermi energy level, indicating that both compounds have metallic properties. The indirect band nature of AlCrF3 is shown by the electronic band structure in Figure 3a, which shows that the conduction group of states minima of AlCrF3 occur at the equilibrium point X and the valence band peak occur at the point M (a).The metallic character of the AlCrF3 combina tion is obvious. Simultaneously the GGA + U were applied for the AlCrF3, due to the tran sition element. The effective Hubbard parameter (Ueff) in the AlCrF3 system was adjusted to 4 eV for the Cr-3d orbital. That shows that the nature of the material changed from metallic to half metallic nature (see Figure 3b). The minima of conduction band of AlBF and the maxima of valence band occur at R of Brillion Zone, which demonstrates the direc

Electronic Properties
In this unit, the energy bands structures and DOS of compounds under investigation are reported. In the primary Brillion area of AlMF 3 (M=Cr, B), the estimated bands structure and symmetry directions are shown in Figure 3. Since no zero energy should exist at or close to the maxima location of the valence band. Using the said potential approach, the valence band maxima of the compound AlBF 3 overlap the minima of conduction band and zero band gap while the band gaps for the compounds AlCrF 3 were determined to be 0.71 eV, respectively. The valence band maximum in both cases crosses the fermi energy level, indicating that both compounds have metallic properties. The indirect band nature of AlCrF 3 is shown by the electronic band structure in Figure 3a, which shows that the conduction group of states minima of AlCrF 3 occur at the equilibrium point X and the valence band peak occur at the point M (a).The metallic character of the AlCrF 3 combination is obvious. Simultaneously the GGA + U were applied for the AlCrF 3 , due to the transition element. The effective Hubbard parameter (U eff ) in the AlCrF 3 system was adjusted to 4 eV for the Cr-3d orbital. That shows that the nature of the material changed from metallic to half metallic nature (see Figure 3b). The minima of conduction band of AlBF 3 and the maxima of valence band occur at R of Brillion Zone, which demonstrates the direct band gap. The energy band structures and DOS of the compounds of interest are described in this section. Figure 3 shows the computed AlMF 3 (M=Cr, B) band structure, as well as the symmetries orientations within the equilibrium geometry within the first Brillion region. in this section. Figure 3 shows the computed AlMF3 (M=Cr, B) band structure, as well as the symmetries orientations within the equilibrium geometry within the first Brillion region.   Figure 4a depicts the electronic states contributions using the T-DOS and P-DOS (total density and partly density of states) to the VB and CB bands for both compounds under inspection. As depicted as EF, the vertical dotted line represents Fermi level. The left-hand side of Fermi levels is valance band while on the right of the Fermi level is conduction band. The various states of the constituent materials are used to gain a better understanding. The plots of total DOS and partial DOS for AlCrF3 show that the VB is governed by  Figure 4a depicts the electronic states contributions using the T-DOS and P-DOS (total density and partly density of states) to the VB and CB bands for both compounds under inspection. As depicted as EF, the vertical dotted line represents Fermi level. The left-hand side of Fermi levels is valance band while on the right of the Fermi level is conduction band. The various states of the constituent materials are used to gain a better understanding. The plots of total DOS and partial DOS for AlCrF 3 show that the VB is governed by Al-tot, Cr-tot and F-tot due to the superposition of all states, but F-s, Cr-s, Cr-p, Al-s, Al-p states do not contribute. The Al and Cr element plays a crucial role in CB while in the valence band the most impact comes from the elements F and Al. In case of AlBF 3 , the VB is led by F-p states, with Al-s and Al-p states contributing only minimally. In conductional band for the compound AlBF 3 most contribution come from the states Al-p and B-p while F-p and B-s take part only in small amount. Al-tot, Cr-tot and F-tot due to the superposition of all states, but F-s, Cr-s, Cr-p, Al-s, Alp states do not contribute. The Al and Cr element plays a crucial role in CB while in the valence band the most impact comes from the elements F and Al. In case of AlBF3, the VB is led by F-p states, with Al-s and Al-p states contributing only minimally. In conductional band for the compound AlBF3 most contribution come from the states Al-p and B-p while F-p and B-s take part only in small amount.

The DOS (Density of States)
(a) TDOS and PDOS for the stress-free AlCrF3 compound with and without incorporating Hubbard correction U were performed in order to examine the influence of the Hubbard U constraint on the stress-free electronic structure ( Figure 4b). The spin of majority states are pushed up close to the FL when Hubbard correction is used, but the spin of minority states become less localized close to and above the FL. As a result, utilizing GGA+U, the AlCrF3 structure becomes half metallic.
Although Cr atoms exhibit net up and down magnetic moments, as shown in Figure  2b,c, it is important to keep in mind that, close to the Fermi level, the Cr atoms mostly TDOS and PDOS for the stress-free AlCrF 3 compound with and without incorporating Hubbard correction U were performed in order to examine the influence of the Hubbard U constraint on the stress-free electronic structure ( Figure 4b). The spin of majority states are pushed up close to the FL when Hubbard correction is used, but the spin of minority states become less localized close to and above the FL. As a result, utilizing GGA+U, the AlCrF3 structure becomes half metallic.
Although Cr atoms exhibit net up and down magnetic moments, as shown in Figure 2b,c, it is important to keep in mind that, close to the Fermi level, the Cr atoms mostly contribute in the valance band in up spin states and in the conduction band in down spin states. The predicted magnetic moment for the Cr atom is around 3.40620 B.

Elastic Properties
Constants of Elasticity Cij, which are important and necessary, can be practiced to characterize elastic properties of materials. The significant properties of the compounds are elastic characteristics which help in finding the stress reaction, applying the macroscopic stress to the chemicals compounds. The elastic constants describe how a material deforms under tension and then returns to its initial shape once the stress has removed [35]. These elastic constants can be used to explain structural stability, atomic bonding properties, and isotropic or anisotropic properties in the atomic sphere. The three different and selfregulating elastic coefficients for a cubic system are C 11 , C 12 , and C 44 . C ij is determined by tilting the unit cell with the proper strain tensor to create an energy strain relation ( Table 1). IRelast-package [36], which Jamal Murtaza advanced, is used and executed inside the WIEN2k code in this work. The ideal cubic crystal structure of every Al based fluoroperovskite in this study as revealed in Figure 1. The relationship between energy and volume deviation is depicted in Figure 2. The Fitting the Birch-Murnaghan equation of states provides the lattice coefficients at equilibrium [37][38][39][40][41]. The reaction caused by the application of force to a crystal using its elastic factors was calculated, which gives a lot of information about the mechanical properties of the material. C 11 , C 12 , and C 44 were used to compute the elastic properties of the cubic symmetric crystals, including their stiffness and stability (three self-governing elastic factors). The computed elastic constant (C ij ) values are shown in Table 2. The experimental values were compared to the elastic constants discovered for AlCrF 3 . The elastic constants for AlCrF 3 (79.11 GPa, 44.85 GPa, 11.86 GPa) and AlBF 3 (77.00 GPa, 94.23 GPa, 31.92 GPa) were computed using DFT.
The study of AlCrF 3 showed that the TB-MBJ approximation potential method provides comprehensive, extra-exact elastic characteristics that are nearer to the experimental results than using the GGA-WC method. Using the calculation, we can obtain the (B) bulk modulus from constants of elasticity using their relationships (ECs). The ECs were all positive except for C 44 and met the criteria. C 11 > 0; C 44 > 0; (C 11 + 2C 12 ) > 0; (C 11 -C 12 ) > 0; C 12 >B > C 11 for flexible stability [37]. Table 2 displays the outcomes of the following relationships when applying Young's modulus (E); the Anisotropy factor (A), Poisson's ratio (v), and Pugh's index ratio (B/G) are obtained.
Fracture resistance is offered by the bulk modulus B, while plastic deformation occurs due to the shear modulus G resistance. The B/G ratio determines the brittleness or ductility of a substance [24]. If the value of B/G is larger than 1.75, it exhibits ductility, and it exhibits brittleness if the B/G is less than 1.75; so, both materials showed ductility because the ratio of B and G (Pugh's criterion) value was greater than 1.75. Poisson's v ratio is a measure of a material's brittleness [25]. However, the materials were both ductile, as illustrated in Table 2, and would be ductile if v was more than 0.26; otherwise, they would be brittle. The elastic anisotropy factor "A." must be equivalent to one; otherwise, the substances would be isotropic. The deviation from one tells us how elastically anisotropic the crystal is, to a certain extent. We report that the value of A for both of our compounds was different from one, indicating that they are all anisotropic. The best indicator of a material's stiffness is Young's modulus (E). For a given substance, when Young's modulus is high, the material is stiffer, and both compounds showed the stiffest property. More than any other elastic attribute, the bonding power as measured by Poisson's ratio reveals the bonding power's uniqueness.

Optical Properties
The optical characteristics of the solid materials were designed using the TB-MBJ approximation potential technique. Characterizing electronics' complex dielectric function for solid materials, i.e., ε(ω), allows us to learn about their optical properties. Both inter-band and intra-band transitions are important in conveying this complex function. Contributions from inter-bands are particularly important for metals [38] and are classified as direct or indirect transitions. Due to their low contribution to ε(ω) and their phonon scattering nature, we neglected indirect inter-band transitions. An electronic system's linear response to the application of an exterior electric field is provided by the anisotropic property of the material with the dielectric function ε(ω), which is just a complex form of the second-order symmetric tensor. Ehrenreich and Cohen's equation define a complex dielectric function that can help to find the ocular properties of a compound [34].
The ε 1 (ω) and ε 2 (ω) are real and non-real components, respectively. The real portion of the dielectric permittivity can be resolved from the imaginary portion, and it only depends on ω 2 -ω 2 , which provides the integral denominator; it also has a direct correlation with the energy gap Eg. By the application of the Kramers-Kroning relationship, the imaginary part can be extracted from the real part as: The real portion ε 1 (ω) can be obtained by using the Kramers-Kronig relationship, given as: For the material response to incident light, the incident optical energy series is 0-14 eV is used. The dispersive effects are demonstrated using a genuine part of the material's surface, whereas light absorption is demonstrated using a model.

Refractive Index
The fictitious region exhibits optical transitions and light absorption within the energy bands. Other optical properties including the absorption coefficient, refractive index, optical conductivity, extinction coefficient, and reflectivity can be calculated using the dielectric functions [28]. The conductivity of the compound AlCrF 3 increased with the energy range 0 to 14 eV, as seen in Figure 5. AlCrF 3 and AlBF 3 had maximal optical conductivity values of 9.8 eV and 6.0 eV, respectively. The ε1(ω) and ε2(ω) are real and non-real components, respectively. The real portion of the dielectric permittivity can be resolved from the imaginary portion, and it only depends on ω′ 2 -ω 2 , which provides the integral denominator; it also has a direct correlation with the energy gap Eg. By the application of the Kramers-Kroning relationship, the imaginary part can be extracted from the real part as: The real portion ε1(ω) can be obtained by using the Kramers-Kronig relationship, given as: For the material response to incident light, the incident optical energy series is 0-14 eV is used. The dispersive effects are demonstrated using a genuine part of the materialʹs surface, whereas light absorption is demonstrated using a model.

Refractive Index
The fictitious region exhibits optical transitions and light absorption within the energy bands. Other optical properties including the absorption coefficient, refractive index, optical conductivity, extinction coefficient, and reflectivity can be calculated using the dielectric functions [28]. The conductivity of the compound AlCrF3 increased with the energy range 0 to 14 eV, as seen in Figure 5. AlCrF3 and AlBF3 had maximal optical conductivity values of 9.8 eV and 6.0 eV, respectively. The refractive index of a material tells us that how fast light flows through the material. Some formulae [29] can be used to calculate the refractive index. We can see in Figure  6 that the refractive index "n" of the compound AlCrF3 had a maximum value of 3.25 at 0.4 Ev and the compound AlBF3 had a maximum value of 3.0 at 0 eV, respectively. The refractive index value for the compound AlCrF3 was 2.75 at 0 eV energy, gradually declining with various peaks. The maximum refraction of the incident light occurred at 0 eV for the compound AlBF3 and then gradually decreased. Different peaks at 3.25, 2.55, 2.0, and 1.5 eV were seen for AlCrF3, whereas AlBF3 had peaks at 3.0, 2.44, 1.6, and 0.5 eV, as shown in Figure 6. The refractive index of a material tells us that how fast light flows through the material. Some formulae [29] can be used to calculate the refractive index. We can see in Figure 6 that the refractive index "n" of the compound AlCrF 3 had a maximum value of 3.25 at 0.4 Ev and the compound AlBF 3 had a maximum value of 3.0 at 0 eV, respectively. The refractive index value for the compound AlCrF 3 was 2.75 at 0 eV energy, gradually declining with various peaks. The maximum refraction of the incident light occurred at 0 eV for the compound AlBF 3 and then gradually decreased. Different peaks at 3.25, 2.55, 2.0, and 1.5 eV were seen for AlCrF 3 , whereas AlBF 3 had peaks at 3.0, 2.44, 1.6, and 0.5 eV, as shown in Figure 6. As the incident photon energy increased, in both cases, the value of the refractive index decreased at 14 eV; most of the light passed through the material and it acted as transparent material, as shown in Figure 6.

Absorption Coefficient
The absorption constant of each substance tells us how these substances would react to radiation. The frequency affects how photons interact with electrons and cause electrons to travel from the valence region to the conduction region grounded on the absorption constant. The absorption coefficient is a measurement of a material's capacity to absorb a specific energy photon [30]. The absorption coefficients support both the real component as well as the imaginary components of the dielectric functions. This is how it is written: The computed absorption coefficients of AlCrF3 and AlBF3 are shown in Figure 7. The absorption section of the AlCrF3 spectra began at 2.0 eV and the peak value was 120 at 10.5 eV, while the absorption coefficient for the compound AlBF3 spectra began at 0.5 eV and peaked at 140.0 at 13.6 eV, as shown in Figure 7.  As the incident photon energy increased, in both cases, the value of the refractive index decreased at 14 eV; most of the light passed through the material and it acted as transparent material, as shown in Figure 6.

Absorption Coefficient
The absorption constant of each substance tells us how these substances would react to radiation. The frequency affects how photons interact with electrons and cause electrons to travel from the valence region to the conduction region grounded on the absorption constant. The absorption coefficient is a measurement of a material's capacity to absorb a specific energy photon [30]. The absorption coefficients support both the real component as well as the imaginary components of the dielectric functions. This is how it is written: The computed absorption coefficients of AlCrF 3 and AlBF 3 are shown in Figure 7. The absorption section of the AlCrF 3 spectra began at 2.0 eV and the peak value was 120 at 10.5 eV, while the absorption coefficient for the compound AlBF 3 spectra began at 0.5 eV and peaked at 140.0 at 13.6 eV, as shown in Figure 7. As the incident photon energy increased, in both cases, the value of the refractive index decreased at 14 eV; most of the light passed through the material and it acted as transparent material, as shown in Figure 6.

Absorption Coefficient
The absorption constant of each substance tells us how these substances would react to radiation. The frequency affects how photons interact with electrons and cause electrons to travel from the valence region to the conduction region grounded on the absorption constant. The absorption coefficient is a measurement of a material's capacity to absorb a specific energy photon [30]. The absorption coefficients support both the real component as well as the imaginary components of the dielectric functions. This is how it is written: The computed absorption coefficients of AlCrF3 and AlBF3 are shown in Figure 7. The absorption section of the AlCrF3 spectra began at 2.0 eV and the peak value was 120 at 10.5 eV, while the absorption coefficient for the compound AlBF3 spectra began at 0.5 eV and peaked at 140.0 at 13.6 eV, as shown in Figure 7.

Reflectivity
Reflectivity is the compound's property that tells us that when light strikes it, it reflects the light back to us. How much of this radiation will hit a compound's surfaces? It will reverse to its former position if it reaches a certain quantity. In optical physics, the term "R(w)" is capitalized to indicate reflectivity. Utilizing the dielectric function's imaginary component, the reflection coefficient was calculated: Reflectivity is the compound's property that tells us that when light strikes it, it reflects the light back to us. How much of this radiation will hit a compound's surfaces? It will reverse to its former position if it reaches a certain quantity. In optical physics, the term "R(w)" is capitalized to indicate reflectivity. Utilizing the dielectric function's imaginary component, the reflection coefficient was calculated: Figure 8 shows that the value of static reflection R(0) for the compound AlCrF3 was almost similar to R(0) = 0.3 at zero photon energy, whereas for AlBF3 it was 0.23. The compound AlBF3 had the highest radiation reflection, with R(w) = 0.34, 0.35, and 0.55 at 0.5 6.3, and 13.5 eV. For the compound AlCrF3, the highest peak reflectivity values were 0.30 at 0.0 eV, 0.34 at 6.3 eV, and 5.5 at 13.5 eV. Figure 8 shows that these materials had lower reflection values. The coefficient is measured in terms of energy. Because these compounds are used by professionals as anti-reflecting coaters, their low reflection coefficient score-especially in the visible and infrared sections-suggests that they appear transparent in these regions. Due to their employment by scientists as anti-reflecting coaters, the small score of this phrase, mostly in the visible and infrared quotas, shows that these substances exhibit transparency in these areas.

Conclusions
The current study used TB-MBJ potential approximations to examine the structural optical, elastic, and electrical features of AlMF3 (M = Cr, B) fluoroperovskites. In general the PBE scheme underestimated the value of the band gap, and it was shown that a U value of 4 eV for Cr atoms produces much better results. The equilibrium lattice constants of the AlCrF3 were found to be between 7.96 and 7.80, whereas those of the AlBF3 were found to be between 4.6999 and 4.70. The elastic properties of the ECs, anisotropy factor bulk modulus, Poisson's ratio, Young's modulus, and Pugh's ratio were all as expected The Pugh (B/G) ratio indicated ductility in the substances under examination; the estimated Poisson ratio values supported the ductility of the material. Both compounds were brittle, stiff, and anisotropic, and they had ionic bonding. AlCrF3 showed indirect band gap behavior at the (X-M) symmetry point according to our observations, but AlBF3 had overlapping of the states, according to our calculations. Using the GGA + U method for AlCrF3 resulted in it having a half-metallic nature. When the calculated results were compared to existing experimental and theoretical data, they were found to be consistent. The coefficient is measured in terms of energy. Because these compounds are used by professionals as anti-reflecting coaters, their low reflection coefficient score-especially in the visible and infrared sections-suggests that they appear transparent in these regions. Due to their employment by scientists as anti-reflecting coaters, the small score of this phrase, mostly in the visible and infrared quotas, shows that these substances exhibit transparency in these areas.

Conclusions
The current study used TB-MBJ potential approximations to examine the structural, optical, elastic, and electrical features of AlMF 3 (M = Cr, B) fluoroperovskites. In general, the PBE scheme underestimated the value of the band gap, and it was shown that a U value of 4 eV for Cr atoms produces much better results. The equilibrium lattice constants of the AlCrF 3 were found to be between 7.96 and 7.80, whereas those of the AlBF 3 were found to be between 4.6999 and 4.70. The elastic properties of the ECs, anisotropy factor, bulk modulus, Poisson's ratio, Young's modulus, and Pugh's ratio were all as expected. The Pugh (B/G) ratio indicated ductility in the substances under examination; the estimated Poisson ratio values supported the ductility of the material. Both compounds were brittle, stiff, and anisotropic, and they had ionic bonding. AlCrF 3 showed indirect band gap behavior at the (X-M) symmetry point according to our observations, but AlBF 3 had overlapping of the states, according to our calculations. Using the GGA + U method for AlCrF 3 resulted in it having a half-metallic nature. When the calculated results were compared to existing experimental and theoretical data, they were found to be consistent.