Ab-Initio study of substitution at alkaline earth site on quaternary Heusler alloys LiXNiSb

In this study, we have studied the quaternary Heusler alloys with the formula LiXNiSb (X = Be, Mg, Ca, Sr, Ba) to investigate the effect of alkaline earth elements at the X site. We explored the structural, lattice dynamics, electronic, and magnetic properties by employing the density functional theory framework in FPLAPW formalism. All the studied alloys are nonmagnetic with no magnetic moment on the Ni atom. LiBeNiSb and LiBaNiSb are dynamically unstable and exhibit imaginary phonon frequencies. The lattice constant of the studied alloys systematically increases with the size of the alkaline earth element, whereas the bulk modulus decreases. Among the stable alloys, LiMgNiSb is metallic, whereas LiCaNiSb and LiSrNiSb are semiconducting with band gap values of 0.43 and 0.34 eV, respectively. The lattice specific heat and optical properties of the semiconducting alloys have also been computed. Our results demonstrate that the studied LiXNiSb alloys can be good candidates for photovoltaic applications.


Introduction
The quaternary Heusler alloys have attracted significant attention of researchers in recent years [1][2][3][4][5][6] due to their interesting electronic structure, making them suitable for applications in areas such as spintronics, thermoelectrics, optoelectronics [7][8][9].These alloys are known for generating fully spin-polarized currents, and their magnetic and electronic properties can be fine-tuned through suitable elemental composition.The general formula of quaternary Heusler alloys is XX'YZ, which crystallizes in F-43m space group [10].Generally, X, X', and Y sites contain transition metals, and the Z site contains the main group element [11][12][13].However, the alloys containing alkali and alkaline earth metals at X and X' sites have also been explored [9,[14][15][16][17][18].Many of these alloys have shown a wide variety of exciting properties.Nag et al examined the LiMgYZ system with Y = Pt, Pd, Au, and Z = Sb, Sn and discovered Dirac spots in the Brillouin zone, resulting in promising thermoelectric properties.Studies on substituting magnetic elements at the Y site in LiMgYSb have also been performed, leading to the ferromagnetic ground state, Weyl points, and spin-dependent transport properties [9].The halfmetallic character has also been predicted for LiSrFeSb and LiBaFeSb systems [19].All these studies make the platform of quaternary Heusler alloys a rich ground for exploring new materials with exotic properties for future energy and electronic device applications.
Continuing the row of exploring quaternary Heusler alloys, in the present manuscript, we have studied the substitution of alkaline earth elements (Be, Mg, Ca, Sr, Ba) at X site in LiXNiSb Heusler alloys, which contain Ni as a magnetic transition element.This study aims to investigate the feasibility of alkaline earth substitution in quaternary Heusler alloy LiXNiSb and their influence on the structural, electronic, magnetic, and optical properties, as well as the lattice dynamics of these alloys.The main objectives of this study are to explore these alloys for: (i) the structural and dynamical stability, (ii) compute the electronic band structure and electronic density of states for stable alloys, (iii) explore the novelty in the electronic structure of these alloys, (iv) possibilities for the magnetic ground state and half-metallic character.Our findings indicate that the tested alloys have fascinating electronic characteristics.The ground state is nonmagnetic, and no net magnetic moment exists on any of these alloys.Moreover, LiMgNiSb is metallic, whereas LiCaNiSb and LiSrNiSb are semiconducting.

Computational methods
All the calculations presented in the study have been carried out within the density-functional theory [20] framework, employing a full-potential linearized augmented plane wave (FP-LAPW) approach using the opensource ELK code [21].Muffin tin radii were assigned as follows: 1.8 bohr for Li and Be, 2.14 bohr for Mg, 2.15 bohr for Ni, 2.35 for Sb, 2.4 bohr for Ca, 2.5 bohr for Sr and 2.8 bohr for Ba.Convergence was ensured across various computational parameters, including the number of k-points and G max .A value of R MT × G max = 7.0 was employed for all calculations.The k-grid convergence test was conducted using the local density approximation approach for a grid of k points ranging from 4×4×4 to k =16×16×16.It was observed that the 10 × 10 × 10 grid size is optimal for computing total energy and other related parameters.Exchange and correlation effects among the electrons were treated using two popular approximations: local density approximation (LDA) [21] and generalized gradient approximation (GGA) [22], with results from both approximations presented for comparison purposes.The lattice dynamics calculations have also been performed for these alloys by employing the supercell approach as implemented in ELK code by employing a 3×3×3 set of Q-points.The optical properties have been computed by employing a 60×60×60 dense grid of k-points in the irreducible Brillouin zone.
The quaternary Heusler alloys possess a cubic crystalline structure with space group F-43 m (No. 216).Within this structure, four distinct Wyckoff sites are available for the basis atoms: 4a (0,0,0), 4b (1/2, 1/2, 1/2), 4c (1/4, 1/4, 1/4), and 4d (3/4, 3/4, 3/4).Three different crystal structures are possible depending on the specific arrangement of atoms occupying these Wyckoff sites.These structures are denoted as type-I, type-II, and type-III, as detailed in table 1.The total energy calculations are critical to ascertain the lowest energy configuration among these structures for a particular arrangement.In the investigated crystal structure of LiAENiSb, Li occupies the X site, alkaline earth (AE) atoms occupy the X' site, while Ni and Sb atoms occupy the Y and Z sites, respectively.

Atomic basis and lattice parameters
The total energy versus lattice parameters for all the investigated alloys for three different arrangements (see table 1) are shown in figure 1.It is evident from all plots that the lowest energy corresponds to the type-III atomic arrangement.This is in contrast to LiAEFeSb [19] or LiMgXY [15] alloys, where all the alloys have a type-I arrangement of the basis as the lowest energy configuration.All the plots for the ferromagnetic arrangement of Ni atoms have been obtained.
There is also a possibility of Ni atoms aligning antiferromagnetics with each other, leading to another lower energy arrangement.To rule out this option, we have also computed the total energy versus lattice constant for these alloys for antiferromagnetic supercells in which the magnetic dipoles of Ni atoms in neighboring cells are aligned opposite to each other.However, we found that the ferromagnetic configuration in the type-III arrangement has the lowest overall energy.
The corresponding lowest energy lattice parameters and bulk modulus values for different alloys are summarized in table 2. We have computed the bulk modulus by employing energy versus volume data and fitted it to different equations of states as implemented in EOS utility in ELK code.Different equations of states employed are (1) Universal equation of state [23], (2) Murnaghan equation of state [24], (3) Birch-Murnaghan 3rd order equation of state [25] (4) Birch-Murnaghan 4th order equation of state [25] (5) Natural strain 3rd order equation of state [26] (6) Natural strain 4th order equation of state [26] (7) Cubic polynomial in V-V 0 .The ( ) 0,0,0 2 results obtained are summarized in table 2. It can be seen that the bulk modulus decreases systematically with an increase in the size of the alkaline earth element.The trends in bulk modulus as obtained for LiAENiSb are similar to that of LiAEFeSb [19], indicating that the alkaline earth ion has a similar impact on the lattice for Fe and Ni.

Lattice dynamics and dynamical stability
Since the alloys we have studied have been predicted computationally, it is essential to establish their dynamical stability [27,28].The dynamical stability of the alloys can be studied by computing phonon dispersions within the supercell approach for the ferromagnetic ground state.The results are plotted in figure 2. It can be seen that LiBeNiSb and LiBaNiSb are not dynamically stable as they exhibit imaginary phonon frequencies around high symmetry point X.LiMgNiSb, LiCaNiSb, and LiSrNiSb do not exhibit any imaginary frequency in phonon dispersion plots and hence can be termed as dynamically stable.The dynamical instability of beryllium (being the smallest) and barium (being the biggest) based alloys seems to be due to the highest chemical stress on the unit cell, which might lead to dynamical instability.For further calculations, we have proceeded with dynamically stable alloys, i.e., LiMgNiSb, LiCaNiSb and LiSrNiSb.Moreover, the phonon cut-off frequency for LiBeNiFeSb is highest, which lowers with an increase in the size of the alkaline earth ion.There are three broad bands in the phonon dispersions; the first is the acoustical branch, followed by two well-separated bands of optical branches.

Electronic properties
For the dynamically stable alloys LiMgNiSb, LiCaNiSb, and LiSrNiSb, we have computed the electronic band structure and density of states, and the plots are shown in figure 3. The first thing to notice is that in spinpolarised electronic band structures, the up and down spin states are almost degenerate.Due to this, the unit cell has no magnetic moment despite the spin polarization.Since Sr in LiSrNiSb has a high atomic number, spinorbit coupling significantly impacts the electronic band structure.This is also visible in the splitting of bands LiSrNiSb band structure, as shown in figure 3.However, the splitting is not very prominent in the conduction band minima or valance band maxima.
It is interesting to note that the alloys have varied electronic properties.LiMgNiSb is metallic, with bands crossing the Fermi level and low density of states at the Fermi level.LiCaNiSb and LiSrNiSb possess a band gap at the Fermi level.The band gap for LiCaNiSb is a direct gap of 0.43eV along the Γ-K direction near Γ point.For LiSrNiSb, the band gap is an indirect gap of 0.34eV with conduction band minima at Γ point and valance band maxima along Γ-K direction near Γ point.Moreover, despite having magnetic Ni in the unit cell, these alloys are nonmagnetic and do not have a net magnetic moment.

Lattice specific heat
The phonon dispersions of a material also contribute to its phononic-specific heat.Therefore, by employing the phonon dispersions of the studied alloys, we have computed their phononic-specific heat as a function of temperature.The phonon-specific heat plots are depicted in figure 4. The specific heat of the alloys is similar to Debye-like curves, which approach the Dulong-Petit limit at high temperatures.Moreover, the specific heat of LiSrNiSb is slightly higher than LiMgNiSb, which is higher than LiCaNiSb.

Optical properties
We have also computed the optical properties for the semiconducting alloys LiCaNiSb and LiSrNiSb within random phase approximation (RPA).The bandgap in density functional theory is underestimated significantly, which poses a challenge for calculating correct optical properties.Scissor correction [29] has been employed to correct the LDA band structure by numerically shifting the conduction band to enhance the band gap value.This results in improved optical properties of the materials.
The dielectric constant of any material consists of real and imaginary parts, which are plotted in figure 5.The real part of dielectric function is related to the ability of a material to store electric energy and the imaginary part is related to absorption by the material when subjected to electromagnetic radiation.The values of static dielectric function ε real (0) are ∼21 and ∼23 for LiCaNiSb and LiSrNiSb, respectively.The highest peak for ε real (max) is around 1eV for LiCaNiSb, and for LiSrNiSb, it is at slightly lower energy, which can be related to their bandgap values.
The imaginary parts for both LiCaNiSb and LiSrNiSb have a sharp onset peak around 0.9eV, which is followed by a broad band of absorption up to a second peak around 3.2eV for LiCaNiSb and 2.9eV for LiSrNiSb.From imaginary parts of the dielectric function for both alloys, it can be seen that there is significant absorption in the visible region, making them a suitable choice for solar cell applications.

Conclusions
In conclusion, we have studied the quaternary Heusler alloys LiXNiSb for isoelectronic substitution at X with alkaline earth elements.It was found that all the alloys prefer to crystallize in type-III arrangements of basis, which is in contrast with other quaternary Heusler alloys.Moreover, these alloys do not possess a resultant magnetic moment in the unit cell despite the magnetic Ni.The magnetic moment on Ni is zero in these alloys.Out of all studied alloys, LiBeNiSb and LiBaNiSb are dynamically unstable, whereas LiMgNiSb, LiCaNiSb, and LiSrNiSb are stable.The stable alloys' electronic band structure and density of state reveal that LiMgNiSb is metallic, whereas LiCaNiSb and LiSrNiSb are semiconducting.The dielectric function of LiCaNiSb and LiSrNiSb suggests that these alloys have significant absorption in visible regions, making them promising candidates for solar cell applications.

Data availability statement
The data cannot be made publicly available upon publication because no suitable repository exists for hosting data in this field of study.The data that support the findings of this study are available upon reasonable request from the authors.

Figure 2 .
Figure 2. Phonon dispersion plots for the studied alloys as computed using supercell method within local density approximation.

Figure 3 .
Figure 3. Spin-polarised electronic band structure and density of states for the dynamically stable alloys.

Figure 4 .
Figure 4. Temperature dependence of phononic specific heat for the studied alloys.

Figure 5 .
Figure 5. Optical properties of LiCaNiSb and LiSrNiSb computed with random phase approximation.

Table 1 .
The different potential crystal structures of quaternary Heusler alloys.

Table 2 .
Lattice parameters and bulk modulus of studied alloys.