First Principles Study of Structural , Elastic , Electronic and Optical Features of the Non-centrosymmetric Superconductors SrMGe 3 ( Where M = Ir , Pt , and Pd )

BaNiSn3-type superconductors SrIrGe3, SrPdGe3 and SrPtGe3 have the critical temperature of 1.80 K, 1.49K and 1.0K respectively have been reported recently. Employing the first-principles method based on the density function theory, we have examined the physical properties including structural, elastic, electronic and optical phenomena of all these structures. For all the phases our optimized lattice parameters are well accord to the experimental lattice parameters. The positive elastic constants of these compounds revealed that these superconductors possess the mechanical stability in nature. The values of Pugh’s ratio and Poisson’s ratio ensured the brittle manner of these compounds and anisotropic behavior is ensured by the values of anisotropy factor. The soft nature of all compounds is confirmed by the bulk modulus analysis. The values of Vickers hardness indicate that the rigidity decreased in the order of SrIrGe3>SrPtGe3>SrPdGe3. The overlapping of the conduction band and valence band at Fermi level indicates the zero band gaps and metallic nature of SrIrGe3, SrPdGe3 and SrPtGe3. The chief contribution around the Fermi level arises from Ir-5d, Ge-4s, 4p states for SrIrGe3 and Ge-4s, 4p states for SrPdGe3 and Pt-5d, Ge-4s, 4p for SrPtGe3 compound. The study of DOS, Mulliken atomic populations and charge density ensured the existing of complex bonding in SrIrGe3, SrPdGe3 and SrPtGe3 with ionic, covalent and metallic characteristics. The analysis of the dielectric function also ensured the metallic behavior of all these compounds.


INTRODUCTION
The noncentrosymmetric superconductors have attracted much consideration having their unique superconducting properties caused by the Rashba-type antysymmetric spin-orbit coupling which leads to a mixing of the spin-singlet and spin-triplet states [1]. The investigations of specific conditions for superconductivity (SC) in non-centrosymmetric compounds were induced in 2004 by the discovery of UniversePG l www.universepg.com different magnetic properties [6][7][8][9][10][11]. Moreover, CeIrGe 3 [12], CeIrSi 3 [13-15] CeCoGe 3 [16,17] and CeRhSi 3 [18, 19] with BaNiSn 3 -type structure have been found to exhibit pressure induced superconductivity. This finding is very interesting because along the c-axis their structure lacks inversion symmetry. These Ce-based noncentrosymmetric superconductors (NCS) are placed close to a magnetic quantum critical point, making it difficult to show the effects of ASOC and inversion symmetry breaking on superconductivity.
In order to analysis the impact of inversion symmetry breaking on superconductivity, nonmagnetic Rashbatype NCS must be discovered and studied because the extra complications that originate from strong felectron correlations can be prohibited. Among compounds adopting the tetragonal BaNiSn3-type structure [20], the phenomenon of superconductivity has been reported for LaPdSi 3  In the current study, we therefore make a plan to investigate the physical properties including structural properties, elastic properties, electronic properties, optical properties of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 compounds. We have used the density functional theory (DFT) based on CASTEP computer program to discuss the detailed physical characteristics of these compounds. The remaining parts of this research work are organized as follows: the computation detail is given in second section then the result and discussion are given and finally the summary of our study is given.
The parameters for the geometry optimization convergence criteria were imputed at 1.0 × 10 -5 eV/atom for the total energy, 0.03 eV/Å for maximum force, 0.05GPa for maximum stress and 0.001 Å for maximum displacement. The elastic stiffness constants are attained using the stress-strain method [35].

RESULTS AND DISCUSSION
3.1 Structural Properties -All the three noncentrosymmetric SrMGe 3 (M= Ir, Pt, and Pd) compounds investigated here belong to BaNiSn3-type tetragonal crystal structure with the space group I4/mmm (139). Each primitive cell unit cell contains one Sr atom at the 2a(0.00, 0.00, 0.00) position, one M atom at 2a(0.00, 0.00, ZT), one Ge 1 atom at 4b(0.00, 0.50, ZGe 1 ) and two Ge 2 atom at the 2a(0.00, 0.00, ZGe 2 ) sites. This three internal parameters (ZT, ZGe 1 and ZGe 2 ) and two lattice parameters (a and c) characterize the crystal structure of all these compounds. The calculated lattice parameters, tetragonal ratio, volume, bulk modulus and internal parameters are listed in Table 1 for all the investigated NCS with the available experimental values. Here the slight deviation of the optimized lattice parameters UniversePG l www.universepg.com from the experimental values ensures the accuracy of our DFT based calculations. However in some cases we have observed that the optimized lattice parameters are slightly greater than the experimental values which happened due to the over estimation of the GGA based calculations. Three dimensional primitive unit cell.

Elastic constants and mechanical properties -
The elastic constants of any material are strongly correlated with the long-wavelength phonon spectrum; in this manner the elastic properties of super conducting material must be executed [37]. The important information about the dynamic features of crystalline materials is also provided by the elastic constants. The material's stability, ductility, brittleness, anisotropy, stiffness behavior and bonding nature in atom are obtained from the study of mechanical properties. According to the Hook's law, the elastic constants were carrying on from a linear fit of the evaluated stress-strain function [38]. The calculated elastic constants of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 superconductors are represent in Table 2. For tetragonal phase, the elastic constants need to content C 11 > 0; C 33 >0; C 66 >0; C 44 >0 C 11 + C 12 -2C 13 >0; C 11 -C 12 >0 (1) 2(C 11 + C 12 ) + 4C 13 + C 33 >0 We have listed the observed elastic constants for SrIrGe 3 , SrPdGe 3 and SrPtGe 3 superconductors in Table 2. From below Table 2 we can see that the observed values are positive and gratified the above criteria. Hence we can say that the SrIrGe 3 , SrPdGe 3 and SrPtGe 3 superconductors are mechanically stable in nature. It is seen that C 11 is significantly smaller than C 33 , indicating that the chemical bonding strength in the (100) and (010) directions is significantly weaker than the bonding strength in the (001) direction. The value of C 44 is obviously smaller than C 66 , which demonstrates that it is easier for shear deformation to occur along the (001) direction in comparison with the (010) direction.
Where 2 = ( 11 + 12 ) 33 − 2 13 2 And = 11 + 12 + 2 33 − 4 13 We have calculated the Poisson's ratio (v) and the Young's modulus (E) using the following equations, The calculated values of Bulk modulus B, Shear modulus G, Young modulus E, B/G and for the compounds SrMGe 3 (M = Ir, Pd and Pt) are listed in Table 3. It has been seen from Table 3 that the values of B of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 are less than 100 GPa [42] indicating that these are relatively soft materials. The stiffness properties of a compound can be described by Young modulus E. The larger value of E signifies the more stiffness of a compound [43]. These compounds also show the larger bulk modulus than the shear modulus expressing the limitation of mechanical stability for these compounds by [44].
Generally, it is extremely helpful to predict the type of bonding force which makes a solid to reveal ductility or brittleness behavior. The shear modulus is denoted by the resistance to plastic deformation and bulk modulus is defined by the resistance to fracture so that the flexibility of a material is observed by the well-known ratio / called as Pough' ratio [43]. The ductile compound processes the larger value of ⁄ (> 1.75) else the compound will be brittle. The lower value of Poisson' ratio ( < 0.26) shows the brittleness properties of a compound and for any other values the compound will be ductile. According to these conditions SrIrGe 3 , SrPdGe 3 and SrPtGe 3 process the brittleness manner. The universal anisotropy index U can be evaluated by using the following equation [45]: If A U = 0 the crystal is entirely isotropic and any deviation from this value represent the degree of anisotropy in the crystal. According to the values of A U exhibited in Table 3 our studied compounds show anisotropic behavior. It is also obvious that SrPdGe 3 is more anisotropic among them. The Vickers hardness which is also an important mechanical property of a material is obtained by the following equation proposed by Chen et al. [46].
In Table 3 the values of Vickers hardness are tabulated. It is evident from Table 3 that SrIrGe 3 , SrPdGe 3 and SrPtGe 3 are relatively soft materials which are contradicted by softness/hardness characteristics presented by the bulk modulus B.

Electronic Properties and Chemical Bonding -
The electronic band structure, partial density of states (PDOS) and total density of states (TDOS) of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 have been studied and discussed to gain the deep insights into the electronic properties of these superconductors. The Fermi level between conduction band and valance band is indicated in diagram along with the range of total band structure. The electronic band structure diagrams for these compounds are depictured in Fig 2. In these diagrams we can see that the valance band and conduction band are overlapped at Fermi level (E F ) and there is no band gap appeared at E F . Since there is no band gap it can be implies that these compounds under study shows metallic behavior and the metallic nature of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 implies that these compound might be superconductor [47]. The densities of states (partial and total) of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 compounds are plotted on Fig 3. The lower valance bands for SrIrGe 3 (-17.59eV to -16.38eV) and for SrPdGe 3 (-18.73eV to -17.57eV) are consists from Sr-4p state which is dominant for these compounds and for SrPtGe3 (-18.07eV to -16.83eV) consist from Sr-4s, 5s state. The middle valance band for SrIrGe 3 (to be found at -11.64eV to -6.49eV), for SrPdGe 3 (to be found at -12.48eV to -7.13eV) and for SrPtGe 3 (to be found at 12.20eV to 6.68eV) are made up from Ge-4s state.
For SrIrGe 3 the upper valance band (to be found at -5.72eV to 0eV) is mainly originates from Ir-5d, Ge-4p states and Pd-4d, Ge-4p for SrPdGe 3 (to be found at -6.24eV to 0eV).For SrPtGe 3 the upper valance band (to be found at -6.36eV to 0eV) is mainly originates from Pt-5d and Ge-4p states. The contribution of Ge-4p states is dominant for all compounds. The conduction band mainly contributed from Ir-5d and Ge-4s, 4p states in case of SrIrGe 3 and Ge-4s, 4p state for SrPdGe 3 and SrPtGe 3 whereas Ge-4s, 4p orbital is dominant for all compounds. At the Fermi level mainly contribution comes from Ir-5d, Ge-4s, 4p states for SrIrGe 3 and Ge-4s, 4p states for SrPdGe 3 and Pt-5d, Ge-4s,4p for SrPtGe 3 compounds.  The calculated values of density of states at the Fermi level are 1.96eV states eV -1 fu -1 , 2.39eV states eV -1 fu -1 and 1.52 states eV -1 fu -1 for SrIrGe 3 , SrPdGe 3 and SrPtGe 3 respectively. Which type of bond exists in these compounds is clearly known by study of Mulliken atomic population. Here we have study The Mulliken atomic population of these compounds and presented in Table 4. From    Hence from the overall detailed study of DOS, Mulliken atomic population and total charge density of SrMGe3 (M=Ir, Pd, Pt) superconductors we can conclude that all compounds have ionic, covalent and metallic bonds which is the common characteristics of BaNiSn 3 structured compounds. UniversePG l www.universepg.com 3.4 Optical properties -Using the frequency dependent dielectric function ε(ω) = ε 1 (ω)+iε 2 (ω), the optical properties of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 superconductors have been studied. The investigation of the optical function of solids provides excessive information of the electronic properties. From the momentum matrix elements between the unfilled and filled electronic states the imaginary part, ε 2 (ω) of dielectric function can be obtain [50]. This is express by the following function, Where represent the polarization of the incident electric field, ω as the frequency of light, Ω represent the until cell volume, , e is define as the charge of electron, |ѱ | and |ѱ | represent respectively the conduction band wave function and valance band wave function at K. By using Kramers-Kronig transformation the real part ε 1 (ω) can be obtained from the value of ε 2 (ω). The optical properties such as absorption spectrum, loss function, conductivity, dielectric function, reflectivity and refractive index are evaluated by eqs (49)-(54) in ref [51].
The absorption spectra offer useful information about the maximum solar energy exchange efficiency and it show how far light of specific wavelength is passes through a material before being absorbed. The absorption spectra of SrMGe3 (M=Ir, Pd, Pt) are shown in Fig 5(a). Fig 5(a) shows the absorption coefficients of all the phases which begin at 0 eV due to their metallic nature. It has been seen that the nature of absorption curves are almost same for these compounds. Two strong peaks (absorption) are found in the visible and ultraviolet regions for all phases at different energy ranges. These peaks are weak in the visible region but continuously increase in the ultraviolet region and reach maximum value at 9.00 eV. According to this outcome we can say that SrIrGe 3 , SrPdGe 3 and SrPtGe 3 compounds are promising for absorbing materials in the UV region. All the compounds show rather good absorption coefficient in the 9.0 eV to 23.85 eV regions. The spectra of reflectivity of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 are shown in Fig 5(b). Reflectivity is a function of incident light energy and is a measure of the ability of a surface to reflect radiation incident on it. It is attained by the ratio of the energy of the wave reflected from a surface to the energy of the wave incident on the surface [52]. From Fig 5(b) we can see that the reflectivity starts from the value of 0.60 for SrIrGe 3 , 0.98 for SrPdGe 3 (it is also the maximum value) and 0.59 for SrPtGe 3 with zero photon energy. The maximum value of reflectivity appears at 12.36 eV energy is 0.69 for SrIrGe 3 compound, at 13.45 eV energy is 0.69 for SrPtGe 3 compound. It is also evident that all phases can be used as excellent coating materials in the energy range 1.5eV to 14.48 eV.
The reflectivity of these compounds is much higher in the ultraviolet and IR regions. Therefore all the compounds, with roughly similar reflectivity spectra, show good promise as good coating materials in the ultraviolet and infrared regions. The conductivity spectrum of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 are shown in Fig 5(c). The conductivity is an optoelectronic phenomenon in which electrical conductivity of a material rises as a result of absorbing of photons. It helps us to mark out the material will be semiconductor, conductor or superconductor. The investigated conductivity spectra with photon energy of SrMGe 3 (M=Ir, Pd, Pt) are shown in Fig 5(c). The photoconductivity starts with zero photon energy due to the reason that the materials have no band gap which is apparent from band structure signifying the metallic behaviors of these phases. The photoconductivity is maxima at 5.18 eV for SrIrGe3 and 2.84 eV for SrPdGe 3 and SrPtGe 3 compound. No photoconductivity occurs above 26.19 eV.
The energy loss function is defined as the energy loss of a fast electron when it traverses in the material [53]. The frequency at which maximum energy loss happened is known as the Bulk plasma frequency ω p of the material which emerges at ε 1 (ω) =0 and ε 2 (ω) is less than one [54,55]. The energy loss spectra for all these three compounds under investigation are plotted in Fig 5(d). The loss function is maxima at 14.90 eV for SrIrGe 3 and SrPtGe 3 compound and 12.44 eV for SrPdGe 3 compound. These materials become transparent when the plasma frequency is lower than that of incident frequency. UniversePG l www.universepg.com Dielectric function is a crucial factor to know the energy loss and polarizability of a material, while electromagnetic wave passes through it. The real and imaginary parts of dielectric function are shown in Fig  5(e) for SrIrGe 3 , SrPdGe 3 and SrPtGe 3 compounds. It is obvious from the study of chemical bonding and electronic structure that these compounds show metallic behavior in nature. Hence it is necessary to include the Drude term to the dielectric function [53, 56, and 57].
The unscreened plasma frequency 3 eV and damping (relaxation energy) 0.05 eV have been used in the Drude term. Despite some variation in heights and position of peaks, the overall features of our calculated optical spectra of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 are almost similar. It has been observed that for all the phases the real part ε 1 (ω) of the dielectric function became zero at around 0.16 eV, which corresponds to the energy at which the absorption coefficients nearly zero (Fig 5a), reflectivity shows a sharp drop (Fig 5b) and the conductivity (Fig 5c) increases sharply. The large negative value ε 1 (ω) of dielectric constant exhibit the Drude-like behaviors which is common feature for metallic system From Fig 5(e) we have observed that real part of the dielectric function comes to zero from below and the imaginary part of the dielectric function comes to zero from above which also ensure the metallic nature of these compounds.
When light is entered into a material then it is refracted or bent. So how much light is refracted or bent when it traversing through a material, this quantity is measured by a dimensionless parameter called the refractive index [58]. The idea of refractive index of an optical material is very effective for its use in optical tools such as waveguides, photonic crystals, etc.
The refractive indices in terms of real and imaginary of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 are displayed in Fig  5(f). The imaginary part describes the amount of absorption loss and the real part signifies the phase velocity of electromagnetic wave when propagates throughout the material. For all superconductors the static refractive index n (0) is found to have the value 104. UniversePG l www.universepg.com

COCLUSION
In this research work, we have performed the detailed physical properties including structural, elastic, electronic, chemical bonding and optical properties of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 by using CASTEP code based on the density functional theory. The optimize lattice parameters have a slight variation from available experimental data for all compounds. The studies of Pugh's ratio values revealed that all compounds are brittle in nature and the value of Poisson's ratio suggests that central force exists in these compounds. The bulk modulus indicated the soft behavior of SrIrGe 3 , SrPdGe 3 and SrPtGe 3 compounds. The study of elastic constant ensured that all compounds are stable in nature and show anisotropic manner. The band structures and density of states (DOS) revealed the metallic nature of these phases. The chemical bonding analysis ensured the existing of covalent, ionic and metallic bonds in these compounds. The reflection spectra of all the compounds showed that these have the potential to be used as coating material to avoid solar heating up to ~8 eV. The large negative values of real part of the dielectric function revealed the metallic nature of all these compounds. The conductivity spectrum and the absorption coefficient are started from zero energy which also indicated the metallic features of all the compounds.

ACKNOWLEDGEMENT
We would like to thank Department of Physics, Pabna University of Science and Technology, Bangladesh for the condensed matter lab support.

CONFLICTS OF INTEREST
We