Dynamical stability, Vibrational and optical properties of anti-perovskite A3BX (Ti3TlN, Ni3SnN and Co3AlC) phases: a first principles study

We have investigated various physical properties including phonon dispersion, thermodynamic parameters, optical constants, Fermi surface, Mulliken bond population, theoretical Vickers hardness and damage tolerance of anti-perovskite A3BX phases for the first time by employing density functional theory (DFT) methodology based on first principles method. Initially we assessed nine A3BX phases in total and found that only three phases (Ti3TlN, Ni3SnN and Co3AlC) are mechanically and dynamically stable based on analysis of computed elastic constants and phonon dispersion along with phonon density of states. We revisited the structural, elastic and electronic properties of the compounds to judge the reliability of our calculations. Absence of band gap at the Fermi level characterizes the phases under consideration as metallic in nature. The values of Pugh ratio, Poisson ratio and Cauchy factor have predicted the ductile nature associated with strong metallic bonding in these compounds. High temperature feasibility study of the phases has also been performed using the thermodynamic properties, such as the free energy, enthalpy, entropy, heat capacity and Debye temperature. The Vickers hardness of the compounds are estimated to be around 4 GPa which is comparable to many well-known MAX phases, indicating their reasonable hardness and easily machinable nature. The static refractive index n(zero) has been found around 8.0 for the phases under study that appeals as potential candidate to design optoelectronics appliances. The reflectivity is found above 44 percent for the Ti3TlN compound in the energy range of 0 to 14.8 eV demonstrating that this material holds significant promise as a coating agent to avoid solar heating.

there is a metallic structural unit in which the strong covalent bonds exist within the A 3 BX (A and B are different metals and X is C or N) formula. Recently, Zhang et al. [21] reported a family of anti-perovskite A 3 BX structure with high stiffness and excellent damage tolerant properties. Within this structure A 3 B acts as metallic box that contribute to achieve ductility and octahedra XA 3 are in corners of the structure that contribute to strong covalent bonding leading to high stiffness. They have studied 126 A 3 BX phases based on the mechanical properties and predicted six hard and four soft compounds which have shown excellent damage tolerant with highly stiff ceramic nature.
The detail theoretical understanding of physical properties of the materials is essential to select compounds to be used in various technological applications. Dynamical stability check is also required before approval of the materials for practical application under external pressure and temperature conditions. Furthermore, the materials' response to high temperature and pressure can be understood by the study of their thermodynamic behavior that is considered as the root criteria for many industrial applications [22]. Additionally, investigations of optical properties such as reflectivity, absorption coefficient and refractive index of the materials are crucial to predict the suitability of those to be used as solar reflector, solar absorber, coating agent to reduce solar heating and possible optoelectronic maximum stress of 0.02 GPa. The elastic constant tensors, C ij have been calculated by the 'stress-strain' method in-built in the CASTEP program that is also used to evaluate bulk modulus B and shear modulus G of the compounds of interest.

Structural properties
The structure of A 3 BX phases crystallizes with the space group Pm-3m (221) belonging to the cubic system [31]. The unit cell structure of A 3 BX [Ti3TlN (TTN), Ni3SnN (NSN), and Co 3 AlC (CAC)] compounds is illustrated in Fig. 1. There are five atoms in the unit cell with the Wyckoff positions of the A atoms at (0, 0.5, 0.5), B atoms at (0, 0, 0) and X atoms at (0.5, 0.5, 0.5), respectively [32], as displayed in Fig. 1. The optimization of the equilibrium crystal structures of TTN, NSN and CAC are achieved by minimizing the total energy and the structural parameters of the compounds are given in Table 1. Our calculated results are compared with available theoretical and experimental results as listed in Table 1, which affirms the reliability and accuracy of our calculations. The obtained lattice constants a and cell volume of the compounds TTN, NSN and CAC are deviated less than 1.7% from available reported values (Table 1) and offer a very good validation of the earlier theoretical estimates [21,[33][34][35].

Mechanical properties
The study of mechanical properties such as stability, elastic moduli, stiffness, brittleness, ductility, and elastic anisotropy of a material provides with fundamental information required for selected engineering applications. The linear finite strain-stress method within the CASTEP code [36] has been used to calculate the elastic constants. The single crystal elastic constants C ij and various polycrystalline elastic moduli of the compounds are represented in  The compounds TTN and NSN show exceptionally low C 44 value that are related to shear deformation and damage tolerant behavior, while higher C 44 observed for the CAC results in higher shear modulus (G) as shown in Table 2. The shear anisotropy factor (Zener ratio) A, is defined by [38] This index designates the anisotropic nature of the compounds with the possibility of appearance of microcracks. The value of A is unity for completely isotropic material and any deviation from unity (smaller or greater) denotes the degree of anisotropy. The computed values of A ( shear modulus (G) and Poisson's ratio and plotted them in 3D contour plots and in their 2D projections [39]. The plots are almost similar and identical for the compounds under investigation. Therefore, only plots for the TTN compound have been displayed in Fig. 2 as representative. It is clear that the values of Y, G and  are deviated from the spherical shape in 3D and from circles in 2D, indicating the anisotropic nature of the compound. The ratios of maximum and minimum values of Y, G and  define the anisotropy factor (AF) in this particular scheme [39] that signifies the degree of anisotropy and are represented in Table 3.
It is seen from the Table 3 that the AF is almost equal for the TTN and NSN compounds, however it is a bit higher for the CAC compound due to its higher elastic moduli compared to the other two members.
The single crystal elastic constants, C ij and respective compliance tensors S ij (S ij = C ij -1 ) have been used to evaluate the various elastic moduli (B, G, Y, and ν) using the following  [40,41]. These values are given in Table 2  It can be anticipated that the CAC compound would be significantly harder than TTN and NSN due to higher values of both the elastic moduli. The Young's modulus (Y) of CAC is also much higher than that of TTN and NSN indicating that CAC is also stiffer than the others. The three factors, Paugh ratio (G/B) [46], Poisson's ratio v and Cauchy pressure are sensible criteria to classify materials either as ductile or as brittle. It is seen from Table 2    The Cauchy pressure (C 12 -C 44 ) [48] can also be used to examine the bonding nature of solids. A negative value indicates that the compounds is dominated by strong directional bonding (covalent) with brittle nature while a positive value endorses the characteristics related to metallic bondings with quasi-ductile nature. Moreover, the high bulk modulus with low shear modulus of the compounds clearly demonstrates the damage tolerant, easily machinable, quasi-ductile and stiff nature of the materials.    This feature will be discussed in the next section.

Charge density distribution mapping and Fermi surface
The electronic charge density distribution mapping in the contour form (in the units of e/Å 3 ) in the (101) (Fig. 4d). For the CAC compound, first (centre, ash color) sheet is spherical in shape along the Γ-M direction, second sheet is octagon shaped in the middle of the cubic plane along the Γ-R direction and third (outer most) sheet is ribbon-like tubes located in each corners in the Γ-R direction (Fig. 4f). In case of NSN, the sheet is hollow cylindrical shaped along Γ-M and Γ-R directions with six holes at cubic structural planes. It is noteworthy that for the NSN phase, several bands are very close to the E F (Fig. 3b) that is well reflected in the Fermi surface topology (Fig. 4e) and it seems another sheet might be attached with crossing sheet along Γ-R direction as shown in Fig. 4e. The Fermi surfaces of the TTN and CAC compounds are formed due to the highly dispersive Ti-3d, Tl-6s, N-2pand Co-3d, Al-3p, C-2p electronic states, respectively. For the NSN compound, it is due to lowly dispersive Ni-3d, Sn-5s and N-2s orbitals.

Vickers Hardness
The hardness provides a significant role for understanding mechanical behavior of the materials and assists one to select them for specific engineering applications. We have calculated the hardness of the studied phases using established formalism [53,54] and presented those in Table 5.
where P  is the Mulliken population of the -type bond, and Ta 2 InC (4.12 MPa) [51].

Phonon dispersion curve
The materials' structural stability and vibrational contribution to the thermodynamic properties such as thermal expansion, Helmholtz free energy and heat capacity can be understood by the study of phonon dispersion curve (PDC) and phonon density of states (PHDOS). The PDC and PHDOS have been calculated of the phases TTN, NSN, and CAC along the high symmetry directions of the Brillouin zone (BZ) using the density functional perturbation theory (DFPT) based linear-response method [55][56][57]. The PDC and PDOS profiles are shown in Fig. 5 (a-f). The dependence of phonon frequency on wave vector (k) for the low frequency acoustic and high frequency optical modes with an energy gap at the edge of the BZ has been observed (Fig. 5 a, Fig.5 a, c, e) compounds. This result is consistent with the mechanically stability of the phases that has been proven using stiffness constants as discussed in Section 3.2. Zero phonon frequency is indicated by the red dashed line.

Thermodynamical properties
Temperature-dependent thermodynamical potential functions namely Helmholtz free energy F, enthalpy E, entropy S, and the phonon specific heat C v and the Debye temperature Θ D at zero pressure have been calculated using quasi-harmonic approximation [58,59] as shown in Figs. 6. The parameters are evaluated in the temperature range of 0-1000 K where harmonic model is assumed to be valid and no phase transitions are expected for the anti-perovskites considered here. The Helmholtz free energy (F) of the three phases is found to decrease gradually with increasing temperature as depicted in Fig. 6(a). This behavior is canonical since thermal disturbance adds to disorder and the entropy of the compounds increase. The difference between internal energy of a system and the amount of unused energy to perform work (represented by the product of entropy, S, of a system and the absolute temperature, T) is called the free energy. The increasing trend of the internal energy (E) with temperature is observed for the phases as expected for compounds in different phases (Fig. 6a). It is noted from Fig. 6 (a) that for the compounds considered, below 100 K, the values of E, F and TS are almost zero. The specific heat at constant volume, C v of the anti-perovskites are calculated and illustrated in Fig. 6b. It is well known that phonon thermal softening happens with increasing temperature, consequently the heat capacity also increases with increasing temperature. The curves reveal that the trend of C v for the phases TTN, NSN and CAC are almost identical; however C v is a bit lower for the CAC phase. The overall trends of C v (T) can be explained by the Debye model. In the low temperature limit, C v exhibits the Debye-T 3 behavior [60]. At high temperature limit, the curves follow Dulong-Petit law where the C v does not depend strongly on the temperature [61].
The temperature dependence of the Debye temperature Θ D has also been calculated using the PHDOS curves as illustrated in Fig. 6(c). The vibration frequency of particles is changed with temperature that is reflected by the Debye temperature, Θ D . Moreover, it is also linked to the bonding strength. The value of Θ D is higher for the CAC compound than that of TTN and NSN which is in consistent with our previous stiffness based discussions in Section 3.2. The value of Θ D is also estimated through the calculation of average sound velocity using the formalism described elsewhere [62] and are represented in Table 6. Table 6: Calculated density ( in gm/cm 3 ), longitudinal, transverse and average sound velocities (v l , v t , and v m , respectively, all in km/s) and Debye temperature ( D in K) of Ti 3 TlN, Ni 3 SnN and Co 3 AlC compounds.

Optical properties
Time dependent perturbations to the electronic ground states are used to define the interaction between the photons and valence and conduction electrons of a system. Electronic transitions occur between occupied states below the E F to unoccupied states above the E F due to the electromagnetic field that shines onto a material with sufficient energy. Therefore, the optical    The higher negative value ε 1 (ω) implies a Drude-like behavior of the materials which is consistent with our discussion in Section 3.3. The imaginary part, ɛ 2 (ω) of the compounds are depicted in Figs. 7b. The ε 1 (ω) shows zero value from below at around 15.5, 21.5 and 22.3 eV (Fig. 7a) and the ε 2 (ω) approaches zero from above at around 19.7, 25.8, and 24.8 eV (Fig.   7b) for the TTN, NSN and CAC compounds, respectively, which indicates further the metallic nature of the studied anti-perovskites.
The frequency dependent refractive index n and the extinction coefficient k have been presented in Fig. 7(c, d). The high value of n of a material is an essential factor to design optoelectronic devices. The static value of n(0) is found to be 8.01, 7.47 and 7.57 with the main peaks appearing at 0.53, 0.56 and 0.6 eV for the phases TTN, NSN and CAC, respectively. The absorption coefficient is related to the extinction coefficient by k ( = 4k/). The sharp peaks are obtained at 2.6, 2.27 and 1.08 eV for TTN, NSN and CAC, respectively. These peaks are characterized by the intra-band transitions of electrons (Fig. 7d).
The energy loss of the light passing through materials can be measured by the absorption coefficient. The energy dependence of absorption coefficient,  have been evaluated and are shown in Fig. 7(e). The value of  is dominant in the ultraviolet (UV) region while it is weak in the infrared (IR) region. It is noted that the value of increases in the direction of UV region and reaches a maximum value at 6.0, 12.2 and 13.9 eV for the TTN, NSN and CAC compounds, respectively. The highest absorption spectrum is found for the CAC material.
The  is high in wide energy range that could be used in optoelectronic devices in both visible and ultraviolet regions. As expected, due to the metallic nature of studied compounds, the photoconductivity starts at zero photon energy as shown in Fig. 7f. The TTN compound exhibits the highest photo-conductivity at 5.4 eV, and for others the maxima showed up at 4.6 eV (NSN) and 6 eV (CAC).
The reflectivity (R) spectra as a function of energy for the studied phases are illustrated in Fig.   7g. It is seen that the value of R starts with a value of 97% for all phases and rises to maximum values of 85% at 12.0 eV, 68% at 18.0 eV and 79% at 19.27 eV for the TTN, NSN, and CAC compounds, respectively. It is reported that a compound with reflectivity ~ 44% in the visible light region would be capable of reducing solar heating [63]. It is noted that the reflectivity is always above 44% in case of TTN compound. For the phases NSN and CAC, the R values lie below 40% in the IR (1.24 meV -1.7 eV) and visible region (1.7 eV -3.3 eV), while in the UV region, it is above 44%. Therefore, the TTN (Ti 3 TlN) compound is a potential candidate for the practical use as a coating material to reduce solar heating. The reflectivity spectra for all phases approach to zero at the incident photon energy range of 24 to 30 eV.
The important feature of loss spectrum can be understood using the dielectric function that defines energy loss of an electron passing through the material. There is no loss observed up to 13 eV for all the studied compounds. The loss peaks appeared in the energy range from 13 to 25 eV. The peaks are prominent in the energy range where the value of  1 = 0 and  2  1.
The loss peaks are associated with the so called bulk plasma frequency  P that are found to be at 14.9, 21.6 and 22 eV for the TTN, NSN and CAC compounds, respectively. The  P also marks the onset of the fast drop of the reflectivity (R) of the materials under consideration, as shown in Fig. 7g.  to mid-UV region is always above 44%. This makes the compound a potential candidate for the practical use as a coating material to minimize solar heating.