Li2CaGeO4 – Wide Band Gap Semiconductor: First Principles Investigation of the Structural, Electronic, Optical and Elastic Properties

The electronic structure and some of its derived properties of Li2CaGeO4 compound have been investigated. The calculations have been performed using the fullpotential linearized augmented plane wave plus local orbitals method and ultra-soft pseudo-potentials . The optimized lattice parameters are found to be ingood accord with experiment. Features such as bulk modulus and its pressure derivative, electronic band structure and density of states are reported. The elastic anisotropy of the crystal is discussed and visualized. Moreover, the optical properties reveal that Li2CaGeO4 compound are suitable candidates for optoelectronic devices in the visible and ultraviolet (UV) regions.

Wide bandgap semiconductors(WBS) are electronic materials in which the energy of the band-to-band electronic transitions exceeds approximately 3 eV. These materials have different kinds of chemical bonds and of crystal lattice structures, but the electronic and optical processes taking place in them have a great deal in common. Diamond, silicon carbide, gallium phosphide, cadmium sulfide, and some other related compounds of the A II B VI type occupy a special place among the widegap semiconductors [10].
At the same time, research on other semiconductors, and especially wide bandgap semiconductors have allowed to fabricate various power devices reliable and performant enough to design high e ciency level converters in order to match applications requirements. Among these wide bandgap materials, SiC is the most advanced from a techno-logical point of view: Schottky diodes are already ommercially available since 2001, JFET and MOSFET will be versy soon [11]. However, only a limited data have been reported on the fundamental properties of Li2CaGeO4 compound. This has motivated us to investigate on the structural, electronic, elastic and optical properties of this material by performing the band structure calculations using the full-potential augmented plane-wave (FP-LAPW) and the pseudopotential plane wave (PP-PW) methods. The remaining of the paper is organized as follows: the theoretical background is described in Section 2. Results are presented and discussed in Section 3. A summary of the results is given in Section 4.

COMPUTATIONAL METHODS
The atomic structure of Li2CaGeO4 compound is known to crystallize in a tetragonal lattice, which has space group I-42m (121), where Li atom occupy 4 d( 0, 0.5, 0.25), Ca at 2b(0, 0, 0.5) sites, Ge on 2a (0, 0, 0) and O at 8i (0.189, 0.189,0.142 of tetragonal unit cell [12]. The crystal structures of Li2CaGeO4 alloy is shown in Fig. 1. The present computations are performed through the FP-LAPW method using DFT as implemented in WIEN2K code [13]. In the study of structural properties, the exchange correlation energy is treated within the GGA method [14]. We have used lmax = 10 for angular momentum expansion and RMTKmax= 8 as a plane wave cut-off with 3000 k points for tetragonal phase. The radii RMT of the muffin tins (MT) are chosen to be approximately proportional to the corresponding ionic radii. The energy between successive iterations is converged to 0.0001 Ry and forces are minimized to 1 mRy Bohr -1 . A dense k-mesh with 10000 k-points was used in the first Brillouin zone to calculate the optical properties.
We have also used the ultra-soft pseudo-potentials of the Vanderbilt-type [14] and the GGA according to GGA-sol approach [15] was already used to calculate structural, electronic, elastic and optiacl properties of Li2CaGeO4. A computer program CASTEP (Cambridge Serial Total Energy Package) [16]. The kinetic cut-off energy for the plane wave expansion is taken to be 500 eV for all cases being considered here. The special kpoint sampling for the integration of first Brillouin zone has been employed by using the Monkhorst-Pack method with 14x14x18 k-points for tetragonal phase. Based on the Broyden Fletcher Goldfarb Shenno (BFGS) [17] minimization technique, the system reached the ground state via self consistent calculation when the total energy is stable to within 5 x 10 -6 eV/atom the force is less than 10 -2 eV/Å. A dense k-mesh with 24x24x29 kpoints was used in the first Brillouin zone to calculate the optical properties.
The optical properties of matter can be described by the complex dielectric function ε(), which represents the linear response of the system to an external electromagnetic field with a small wave vector. It can be expressed as The investigated compound crystallizes in the tetragonal symmetry. This symmetry group has two dominant components of the dielectric tensor. These dielectric functions are The frequency dependent complex dielectric tensor ε2() components are calculated by using the following mathematical expressions [18]: In the above relations, e 2 = 1/m = 1 and ħ= 1 (written in atomic unit) where '' '' is the photon frequency of energy "   ", and ) ( , k p y x n n  represent the x/y-components of the dipolar matrix elements between initial where M implies the principle value of the integral. The optical constants such as refractive index n(w) and the extinction coefficient , are calculated intermsof the real and the imaginary parts of the complex dielectric function as follows [19,20].

RESULTS AND DISCUSSION
The experimental lattice parameters has been optimized using Birch-Murnaghan's [21] equation of state by fitting energy vs cell volume. Figure 1 presents the structural optimization curves obtained by using the FP-LAPW method. This allowed the determination of the equilibrium lattice constants a(Å) and c(Å) and bulk modulus B (GPa) and its first pressure derivative B'. The resulting structural parameters for both methods being considered in the present work are listed in Table 1. The optimal lattice parameters a(Å) and c(Å) obtained by this procedure accords well with the experimental values reported in Ref. [12]. Based on the experimental data, the equilibrium lattice constants for Li2CaGeO4 are best described by PP-PW, compared with the FP-LAPW method.
Nevertheless, both used methods overestimated the lattice constant with respect to experiment.
The calculated constants Cij allow us to obtain the macroscopic mechanical parameters of these compounds, namely their bulk (B) and shear (G) moduli can be computed with the aid of the main approximation Voigt-Reuss-Hill [25], and and and ) ( 2 The values of Young's modulus and Poisson's ratio were also calculated by using the equations :  The evaluated values of the bulk modulus , shear modulus , Young's modulus and Poisson's ratio  of Li2CaGeO4 using the VRH approximations are given in Table 3.
The bulk and shear modulus can be explained as a measure of the resistance to volume and shape changes respectively. As can be seen from Table 3, that the low values of B and G suggest that the Li2CaGeO4 should be soft materials that can be easily machinable. The Young's modulus E assesses the stiffness of the compound: the larger the value of Young's modulus the stiffer will be the compound. On comparing the results on G with those reported for CuMn2InSe4 [28]. The Young's modulus of Li2CaGeO4 is slightly larger than that of CuMn2InSe4 which means that the former is a much harder than the latter. The studied compound has a moderate bulk modulus, indicating that its resistant to compression is moderate, i.e., it has a high compressibility.
The calculated 3D representations of the directional dependence of the Young's modulus (E) and compressibility (β) and their cross sections in the xy and xz planes are shown in Fig. 2. Obviously, the 3D-closed surfaces (cross sections) of E and β exhibit a pronounced deviation from the spherical shape (circular form), indicating a noticeable directional dependence of the Young modulus and compressibility. Thus, one concludes that Li2CaGeO4 is characterized by a pronounced elastic anisotropy.
The investigation of the electronic band structure and total density of states are important because most of the physical properties of solids are related to them. The spinpolarized band structure in its tetragonal structure has been calculated using the FP-LAPW method. Based on the lattice symmetry, the integration paths P(0, 0, 0. The refractive index is a quantity that describes how much light is refracted after entering a material [29].

CONCLUSION
In conclusion, the structural, electronic, elastic and optical properties of tetragonal Li2CaGeO4 were investigated using both PP-PW and FP-LAPW methods. The computed lattice parameters, namely a and c and the bulk modulus and its first pressure derivative were found to be in good accord with data available in the literature.