Ab initio investigations of structural, elastic, electronic and optical properties of the fluoroperovskite TIXF3 (X=Ca, Cd, Hg, and Mg) compounds

Ab initio calculations of Tl-based fluoroperovskite compounds TlXF3 (X = Ca, Cd, Hg, and Mg) are carried out in the framework of Density Functional Theory (DFT). We have investigated their structural, elastic, electronic and optical properties using the full-potential linearized augmented plane wave (FP-LAPW) method. The exchange-correlation potential is examined using the generalized gradient approximation with additional Hubbard-U term for indulging on-site Coulomb interaction (GGA + U). These compounds have been found to be mechanically stable. The elastic properties such as elastic constants, bulk modulus, anisotropy factor, Poisson’s ratio, and Pugh’s ratio are obtained. The calculations of electronic band structure show that the TlCaF3 and TlMgF3 are direct while TlCdF3 and TlHgF3 are indirect band gap materials. The contribution of the different bands was analyzed from the total and partial density of state curves. Calculations of the optical spectra such as the real and imaginary parts of the dielectric function, optical reflectivity, absorption coefficient, optical conductivity, refractive index and extinction coefficients are performed for the energy range of 0 to 30 eV.


Introduction
Compounds having general chemical composition ABF 3 are well known as fluoroperovskites. In ABF 3 , A and B represent cations while F (Fluorine) is an anion. These compounds form an interesting class of materials with mechanically stable crystal structure while showing good electronic behavior having band gap energy ranging from semiconductors to insulators. These compounds have received much attention in recent years due to their technological importance as a lens material in optical lithography, scintillation materials, radiation dosimeters and in semiconductor industry [1][2][3]. Owing to such broad applications, these compounds are widely examined experimentally and computationally by different authors e.g. see in [4].
Flouroperovskites are generally characterized by their large energy band gap [5][6][7]. The wide energy gap of these compounds makes them technologically important. KMgF 3 and BaLiF 3 are used as vacuum-ultra-violet materials for lenses in optical lithography steppers [8,9]. KMgF 3 is also promising as a scintillation material [2] and radiation dosimeter [10] when doped with lanthanide ions Ce and Er. Theoretical study on Ag-based flouoroperovskites, AgMgF 3 and AgZnF 3 , was reported by Murtaza et al [11]. They predicted wide absorption energy range of reported materials make them suitable for different device applications. The optoelectronic properties of Sn-based flouoroperovskites were studied in [12]. These compounds were found to be electronically insulators and were predicted to have Auger-Free luminescence (AFL). Structural, magnetic and optoelectronic behavior of TlMnX 3 (X=Cl, and F) were reported by F Hamioud et al in [13] and optics technology applications were predicted based on optical spectra. A simulation study on TlCdF 3 revealed its insulating nature, having transparency for a wide range of energies and therefore suggesting it suitable for optical applications [14]. Despite interest in the investigation of fluoroperovskite compounds for various applications, to the best of our knowledge, there is a lack of literature on the study of Tl-based flouroperoskites.
A recent trend in the development of thallium based compounds in the field of radiation detection is observed and various studies have been reported [15,16]. The presence of thallium atom in the chemical composition of these compounds increases the effective atomic number that contributes towards better detection efficiency [17]. Moreover, the simple cubic structure of the compounds makes them technologically promising candidate in the applications where single growth is required.
In this study, we aim to give a detailed description of the structural, elastic, electronic and optical properties of TlXF 3 (X=Ca, Cd, Hg, and Mg) compounds using the GGA+U method. This approach in DFT was proposed in the 90s to treat the electronic correlation with the Hubbard-like model [18]. Since then, it has been widely used to investigate the properties of various materials [19][20][21] including halide perovskites [22].
This paper is organized in 4 sections. Section 2 is devoted to the method of calculation, in section 3, results and discussions are presented and finally, section 4 is devoted to the conclusions of the study.
Simulation study for the investigation of structural, elastic, electronic and optical properties is carried out by applying the general framework of density functional theory (DFT). The full potential linearly augmented plane wave (FP-LAPW) technique [23] is implemented by employing WIEN2K code [24]. The effect of on-site Coulomb interactions is considered and calculations are performed in generalized gradient approximation with additional Hubbard-U term for indulging on-site Coulomb interaction (GGA+U) [25]. Structural parameters are investigated by fitting the energy versus volume curve using Murnaghan's equation of state [26]. For this study, the value of RMT is chosen in such a manner that there is no charge leakage from the core and the total energy is ensured. RMT values of 2.5, 2.01, 2.12, 2.18 and 1.85 are used for Tl, Ca, Cd, Hg and Mg while RMT values for F in TlCaF 3 , TlCdF 3 , TlHgF 3 and TlMgF 3 are 2.01, 1.88, 1.94 and 1.85 respectively. The wave function within muffin tin spheres are expanded in spherical harmonics up to lmax=10, while K points are taken to be 1000 and Gmax is 12. The difference of energy between the core and valence band is taken as 6 Ry.

Results and discussion
3.1. Structural and elastic properties All Tl-based flouroperovskite compounds studied in this work have ideal cubic structure as shown in figure1. The variation of total energies with respect to volume is shown in figure 2. The equilibrium lattice constants are obtained by fitting the Birch-Murnaghan equation of state [26]. These values are listed in table 1. In case of TlCdF 3 , our calculated lattice constant (4.49 Å) is in reasonable agreement with the value (4.395 Å ) found experimentally in [27]. Due to unavailability of the experimental results concerning the equilibrium lattice constants of the rest of studied compounds, we support our calculated results on the basis of known lattice  constant of TlMnF 3 [13]. As Tl and F atoms in the unit cell are the same in our studied compound and those reported in the [13], the lattice constant should increase as the size of the third atom increases. The ionic radius of Mn +2 ion is smaller than Mg +2 , Cd +2 , Ca +2 , and Hg +2 , that's why the ionic radii of all our compound have a larger lattice constant values compared to the lattice constant of TlMnF 3 which is reported to be 4.123 Å. Crystal response to applied forces is determined by the elastic constants which provide important information about the mechanical properties of solid materials. For cubic symmetry crystals, three independent elastic constants C 11 , C 12 and C 44 are used to determine the mechanical properties such as the rigidity and stability of material under study. The calculated values of elastic constants C ij are presented in table 1. For TlCdF 3 , the elastic constants were compared with those found experimentally in [28]. Our calculated values for C 11 and C 44 (106.62, 18.09 GPa) match fairly well with reported values (103.6, 18.08 GPa) while value of C 12 (46.89 GPa) is reasonably close to the value (39.6 GPa) found in [28]. The values of elastic constants were also compared with the values of C 11 , C 12 and C 44 (124.7, 39.59 and 11.53 GPa) obtained through DFT using GGA-WC for TlCdF 3 [14]. The comparison shows that the GGA+U approach provides overall more accurate elastic properties, close to the experimental values, as compared to GGA-WC in case of TlCdF 3 .
The bulk modulus B can be calculated from elastic constants using the relation 11 12 All the elastic constants are positive and satisfy the criteria C 11 >0; C 44 >0; (C 11 −C 12 )>0; (C 11 +2C 12 )>0; C 12 <B<C 11 for mechanical stability [29]. Table 1 presents the results of anisotropy factor A, Young's modulus E, Poisson's ratio ν and Pugh's index ratio B/G by using the following relations [30] ( ) The classification of a material as a ductile or brittle can be made on the basis of the B/G ratio [31]. If this ratio is smaller than 1.75, the material shows brittle nature otherwise it shows ductile character. On the basis of Pugh's criteria, all the compounds show ductile nature. Ductility or brittleness of compounds can also be inferred from the poison's ratio v [32].
The material is ductile if the v is greater than 0.26 otherwise it is brittle. The values listed in table 1 again confirmed that all compounds show ductile behavior. The elastic anisotropy factor A for all studied compounds is presented in table 1. For isotropic materials, this factor is equal to 1 and deviation of values from 1 represents a measure of anisotropy possessed by the material. The calculated values of anisotropy factors are listed in table 1. The values for TlCaF 3 , TlCdF 3 and TlHgF 3 are 0.032, 0.605 and 0.553, respectively while that of TlMgF 3 is 1.23. These results clearly reveal that all compounds are anisotropic. The ductile and anisotropic behavior is in agreement with the similar Tl-based flouroperovskite compound TlMnF 3 [13].

Electronic properties
In this section, we are reporting the energy band structure and density of states of compounds under study. The calculated band structure for TlXF 3 (X=Ca, Cd, Hg, and Mg) along with symmetry directions in the first Brillion zone at equilibrium geometry are given in figure 3. The zero energy is set to coincide with the top of the valence band.
The calculated band gap values for all compounds are found to be 4.39, 3.46, 3.05 and 4.40 for TlCaF 3 , TlCdF 3 , TlHgF 3 , TlMgF 3 , respectively as listed in table 2 using GGA+U. Figure 3(a) shows that both conduction band minima and valence band maxima of TlCaF 3 are located at same X symmetry point in Brillion Zone, exhibiting the direct band gap nature. The band structure of TlCdF 3 is presented in figure 3(b) with conduction band minima occurring at Γ symmetry point while valence band maxima lying at M symmetry point showing indirect band nature. The indirect band nature is consistent with previous study on TlCdF 3 [14]. The band gap nature of TlHgF 3 is also indirect as shown in figure 3(c) where conduction band minima and valence  band maxima are located at Γ and M symmetry points respectively. In the case of TlMgF 3 , as shown in figure 3(d), both conduction band minima and valence band maxima are lying at X symmetry point showing direct bandgap nature. A deeper insight into the electronic structure is obtained by studying the total and partial density of states (TDOS and PDOS) as presented in figure 4. The density of states plot for TlCaF 3 shows that the valence band is dominated by F-p state having a minor contribution from Tl-s state. In conduction band, the major contribution is from Tl-s state. In the case of TlCdF 3 , the valence band is populated by F-p and Cd-d states. The conduction band is dominated by states contributed by Cd. Similar to TlCdF 3 , the valence band of TlHgF 3 has a major contribution from F-p and Hg-d states. The conduction band states are consisting of F-p and Hg-d states.
The DOS plot of TlMgF 3 shows that valence band is mainly populated by F-p while significant contribution is also coming from Tl-s state. In the conduction band, Tl-s state is having a major contribution.

Optical properties
All the calculations for optical properties are carried out using the GGA+U approach. The photon energy range for optical response is taken as 0-30 eV. The optical properties of the compound can be described by the complex dielectric function represented by the Ehrenreich and Cohen's equation [33] ( ) ( ) ( ) ( ) e w e w e w = +i 8 1 2 The real part of the dielectric function is calculated by using the equation The real part shows the dispersive behavior from the material's surface and imaginary part shows the absorption of light for the material [34]. The absorption of light found from imaginary part represents the optical transitions between the energy bands [35].The real and imaginary part of dielectric functions are used for the calculation of other optical parameters such as refractive index, extinction coefficient, the absorption coefficient, optical conductivity and reflectivity [36,37]. Figure 5 shows that the compounds are optically more In the figure 6, we see that the refractive index is behaving quite similar to the dielectric function of the materials. It has peaks at about 3 and 8 eV. Two smaller peaks are also observed at about 26 eV for the compound TlCaF 3 . The refractive index and the extinction coefficients show almost similar behavior but extinction coefficient is smaller than the respective refractive index lower in every energy region. These peaks are due to intraband transitions between valance and conduction bands.

Absorption coefficient
The absorption coefficient contains a contribution from both real and imaginary parts of the dielectric function The absorption coefficient for a material depicts the light absorbed per unit length by an optical system [39]. The absorption is the result of interaction between electrons and photons together with interband and intraband transitions. We see that in figure 7, absorption coefficient has several peaks for all the compounds. TlMgF 3 seems to be more active than all other three compounds however TlCaF 3 has a maximum peak at about 26 eV.

Reflectivity
The reflectivity is calculated from the imaginary part of the dielectric function   Figure 8 shows that there is more optical reflectivity in the low energy region than in the high energy. We find several peaks with the highest peak at around 7 eV for the compound TlMgF 3 .

Optical conductivity
The optical conductivity is calculated from the following equation  In the figure 9, we see that the compound TlCaF 3 is more active in the high energy region as compared to all other compounds which are active in the low energy region.

Conclusions
Structural, elastic, electronic and optical properties of TlXF 3 (X=Ca, Cd, Hg, and Mg) are reported in the present work using GGA+U approximations. The equilibrium lattice constants are found to be in the range of 4.15-4.94 Å. The elastic properties such as elastic constants, bulk modulus, anisotropy factor, Poisson's ratio, and Pugh's ratio are predicted. The B/G ratio shows that all the studied compounds are ductile in nature. The ductile nature is also confirmed from the obtained values of the Poisson ratio. Our calculations show that TlCaF 3 and TlMgF 3 have direct bandgap behavior at X-symmetry point, while TlCdF 3 and TlHgF 3 exhibit indirect band nature. The calculated results are compared and found consistent with available experimental and theoretical data. On the basis of simple cubic structure, large effective atomic number due to the presence of thallium and band gap in insulating materials range, these compounds are promising candidates for scintillation detectors.