An ab initio study of structural, elastic and electronic properties of hexagonal MAuGe (M = Lu, Sc) compounds

In this paper, we performed a detailed theoretical study of structural, elastic and electronic properties of two germanides LuAuGe and ScAuGe by means of first-principles calculations using the pseudopotential plane-wave method within the generalized gradient approximation. The crystal lattice parameters and the internal coordinates are in good agreement with the existing experimental and theoretical reports, which proves the reliability of the applied theoretical method. The hydrostatic pressure effect on the structural parameters is shown. The monocrystalline elastic constants were calculated using the stress-strain technique. The calculated elastic constants of the MAuGe (M = Lu, Sc) compounds meet the mechanical stability criteria for hexagonal crystals and these constants were used to analyze the elastic anisotropy of the MAuGe compounds through three different indices. Polycrystalline isotropic elastic moduli, namely bulk modulus, shear modulus, Young's modulus, Poisson's ratio, and the related properties are also estimated using Voigt-Reuss-Hill approximations. Finally, we studied the electronic properties of the considered compounds by calculating their band structures, their densities of states and their electron density distributions.


Introduction
In the recent years, ternary intermetallic compounds MAuGe (M denoting a rare earth element) have received an increased scientific attention due to their fascinating structural variety, exceptional physical properties, and wide range of applications. As a result, many scientific reports were published on their crystal structures and physical properties [1][2][3][4]. Ternary germanides, for example, provide a wider range of interest in magnetic susceptibility, electrical resistivity and specific heat [4]. Note that the physical properties of the MAuGe ternary germanides strongly depend on the nature of the rare earth element M. According to the authors of the reference [1], the compounds ScAuGe and LuAuGe are diamagnetic materials and exhibit remarkable physical properties which make them interesting for possible technological applications.
Pöttgen R. et al. [1] recently reported the experimental preparation of the new germanides MAuGe (M = Lu, Sc) by melting alloys prepared from their atomic constituents in an arc furnace with the subsequent annealing at 1070 K. The crystal structures of MAuGe (M = Lu, Sc) compounds were determined by X-ray diffraction. They show crystal structures derived from the CaIn 2 structure-type by an ordered arrangement of Au and Ge atoms in the In positions. Note that the MAuGe (M = Lu, Sc) have crystal structures similar to that of the LiGaGe compound [5,6]. In addition, the MAuGe compounds crystallize in the P6 3 space group (No. 186) where the Au and Ge atoms form a three-dimensional (3 )-[AuGe] polyanions of elongated tetrahedra in MAuGe [1]. The frowning degree of the [AuGe] hexagonal greately depends on the size of the atom M = (Lu, Sc), so that the lattice constant of the hexagonal lattice increases systematically with the size of the atom M (M = Lu, Sc). Schnelle et al. [4] classify the MAuGe (M = Lu, Sc) as weak diamagnetics, and measurements of their electrical resistivities indicate a metallic behavior for both compounds.
To the best of the authors' knowledge, no theoretical or experimental study of the elastic properties of these compounds has been carried out up till now. Certainly, it is very important to get to know the elastic properties of a material because they provide information on the stability and stiffness of the material against the externally applied stresses. Due to the close relationships of elastic properties with other fundamental physical properties, in the present work we performed detailed ab initio calculations of structural, elastic and electronic properties of MAuGe with M = (Lu, Sc) under the pressure effect. Note that the measurements of elastic and structural parameters under pressure are generally difficult to determine experimentally. Therefore, to know the elastic constants and the evolution of the lattice parameters under the pressure effect is very important in many modern technologies [7]. We hope that the reported results in this article will be useful for further studies or for possible technological applications of the MAuGe germanides.

Computational details
All quantum mechanics calculations were performed using the pseudopotential plane wave (PP-PW) method in the framework of the density functional theory (DFT) as implemented in the CASTEP code (CambridgeSerial Total Energy Package) [8]. In order to calculate the structural parameters and elastic moduli properties, the exchange correlation energy is treated within the generalized gradient approximation GGA-PBEsol as parameterized by Perdew et al. [9]. For all electronic total energy calculations, the valence electrons of the Lu, Sc, Au and Ge pseudo-atoms are described by the Vanderbiltultra-soft pseudopotential [10]. The Lu 4 14 5 6 5 1 6 2 , Au 5 10 6 1 , Ge 4 2 4 2 and Sc 3 2 3 6 3 1 4 2 orbitals are explicitly treated as valence states. The plane-wave basis set was defined by a plane-wave cut-off energy of 400 eV, and the Brillouin zone (BZ) integration was performed over the 5 × 5 × 4 grid size using Monkhorst-Pack scheme [11] for hexagonal structure. The plane-wave basis set and the grid size were chosen after a convergence test in order to ensure a sufficiently accurate converged total energy, thus optimizinng the geometry, computing the elastic constants and the electronic structures of MAuGe. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization technique [12] was used to determine the structural parameters of the equilibrium geometries because the (BFGS) method provides a way to find the lowest energy of the considred cristalline structure. This set of parameters was carried out with the following convergence criteria: a maximum of an ionic Hellmann-Feynman force within 5 × 10 −2 eV/Å, a maximum stress of 10 −1 GPa, a maximum displacement of 2 × 10 −3 Å and a self-consistent convergence of the total energy of 2 × 10 −5 eV/atom. The well-known stress-strain approach [13,14] was used to determine the elastic constants by applying a set of a difined homogeneous deformation with a finite value and by calculating the resulting stresses in the optimized and relaxed structures. The convergences criteria during the relaxation stage of the internal coordinates were chosen as follows: a total energy less than 4 × 10 −6 eV/atom, a converged forces within 10 −2 eV/Å and a maximum ionic displacement of 4 × 10 −4 Å.

Structural properties
The ternary compounds LuAuGe and ScAuGe crystallize in a hexagonal structure type with the space group 6 3 / (No. 186) [1]. The conventional cell of the MAuGe (M = Lu, Sc) germanides contains two chemical formulae ( = 2). Therefore, it contains 6 atoms per unit cell as show in figure 1. The M, Au and Ge atoms occupy the following Wyckoff positions: M=(Lu, Sc): 2 (0, 0, M ), Au: 2 (1/3, 2/3,  , where M , Au and Ge are the internal -coordinates of M=(Lu, Sc), Au and Ge atoms, respectively. Thus, the MAuGe unit cell is characterized by five structural parameters not fixed by symmetry: two lattice constants ( and ) and three internal coordinates ( M , Au and Ge ).
As the first step of our calculations, we used the experimental structural parameters in order to calculate the optimized lattice constants ( and ) and the internal atomic -coordinates at zero pressure. The calculated structural parameters of MAuGe, including the equilibrium lattice constants, 0 and 0 , and the internal coordinates, 0 , Au 0 and Ge 0 , using the PP-PW method within the GGA approximation are shown in table 1 and table 2 in comparison with experimental [1] and theoretical data [4,15].
As can be seen from table 1 and table 2, our calculated equilibrium structural parameters ( 0 , 0 , M , Au and ) are in very good agreement with the existing experimental and theoretical data. The calculated values of the five optimized structural parameters of LuAuGe (ScAuGe) deviate from the measured ones by less than 0.93%, 1.13%, 0.01%, 0.01% and 0.01% (0.93%, 1.13%, 0.01%, 0.01% and 0.01%), respectively. Moreover, the calculated and measured internal atomic coordinates of M, Au and Ge atoms of the cell unit are in very good agreement with the experimental and theoretical values [1,4,15]. This excellent matching is an indication of the capability of this chosen first-principles method to provide confidence for the calculations of the elastic and electronic properties of the titled compounds. To evaluate the hydrostatic pressure effect on the structural properties ( , and ) of MAuGe (M = Lu, Sc), in figure 2 we illustrated in their relative changes / 0 under pressure, where represents , or , and 0 refers to the structural parameter value at zero pressure. The evolution of the relative changes of each parameter in the considered pressure range was well fitted to a second order polynomial [13]:

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The Hexagonal MAuGe (M= Lu, Sc) compounds The expressions describing the relative changes of structural parameters , and are as follows: It can easily be seen from figure 2 that the relative variation of structural parameters decrease when the pressure goes from 0 GPa to 40 GPa. The estimated values of the linear compressibilities of the lattice parameters and are = −2.02 × 10 −3 −1 and = −3.8 × 10 −3 GPa −1 for LuAuGe and = −1.9 × 10 −3 GPa −1 and = −3.75 × 10 −3 GPa −1 for ScAuGe. We also see that for the two materials, is less than which implies that the / 0 ratio decreases faster than the / 0 ratio. Therefore, the MAuGe compounds are relatively more compressible along the -axis than along the -axis. It hould also be noted that the clearly different values of and reveal a notable compression anisotropy. The linear ( and ) and volumic compressibilities obtained from the lattice parameters ( and ) and volume , respectively, were used to estimate the bulk modulus as follows [14]: The calculted values of the bulk modulus using the equations (3.4) and (3.5) are listed in table 3. The bulk modulus can also be extracted from the fit of the data Energy-Volume ( − ) (Pressure-Volume . We can obseve that there is a good agreement between different procedures used to evaluated the bulk modulus . Therefore, these results constitute a good proof of the reliability for our calculations. Calculated from Vinet EOS [20] Calculated from Murnaghan EOS [17] Calculated from Birch-Murnaghan EOS [18] Calculated from Birch EOS [16] Calculated from linear compressibilities and : = 1/(2 + ) Calculated from the volumic compressibility : = 1/ . In order to fully characterize the pressure dependence of the structural parameters, we studied the pressure effect on the bond lengths between the following first atomic neighbours: Ge 1 , Ge 2 , Au 1 , Au 2 , Lu 1 , Lu 2 , Sc 1 and Sc 2 . The pressure effect on the normalized bond lengths / 0 is plotted in figure 5. stands for the bond-length at pressure , and 0 is its corresponding value at zero pressure.

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The Hexagonal MAuGe (M= Lu, Sc) compounds    From figure 5, we observe that a different bond-length as a fuction of pressure in MAuGe compounds perfectly follows the second order polynomial as shown in equations (3.6) and (3.7). The calculated bond lengths at zero pressure are listed in table 4 with the available experimental data [1]. From the reported results in table 4, it can be seen that the Lu 1 -Lu 2 and Sc 1 -Sc 2 bonds are more compressible than the other bonds, while the Ge 1 -Au 2 bond in the two compounds is the least compressible. Note that our results are in good agreement with the reported experimental findings [1]. (3.7)

Single-crystal elastic constants
The elastic constants are important physical parameters for solid crystalline materials. In particular, they provide an information on the response of the material when an external mechanical stress is applied and regarding the nature of the forces acting in solid materials [22]. A crystalline solid material of a hexagonal symmetry is described by five independent elastic constants, namely 11 , 33 , 44 , 12 and 13 . The calculated numerical values of the five elastic constants at zero pressure are listed in table 5 for the LuAuGe and ScAuGe compounds. Note that there are no experimental or theoretical results available in the literature for the elastic constants of the MAuGe compounds (M = Lu, Sc) to be compared with our results. The present work is the first attempt to calculate the elastic constants of the titled compounds.
From the obtained results, we can draw the following conclusions: (i) we observe that 11 is greater than 44 , 12 and 13 , which shows that the considered systems are more resistant to unidirectional compressions than to shear strains; (ii) knowing that 11 and 33 reflect the uniaxial stiffness along the and axes, respectively, the obtained 11 value is greater than that of 33 for the two compounds, indicating that MAuGe (M = Lu, Sc) are relatively stiffer materials along the axis than along axis. This result agrees very well with the results already obtained by adjusting the relative changes of the lattice constants and as a function of pressure illustrated in figure 2; (iii) figure 6 shows the pressure dependence of the five independent elastic constants of the MAuGe compounds (M = Lu, Sc) for pressures up to 40 GPa. It can be clearly seen in figure 6 that the elastic constants increases monotonously with the increasing pressure, and the fit results are given in equations (3.9) and (3.10). (iv) The mechanical stability of MAuGe compounds (M = Lu, Sc) is verified because the calculated at zero pressure satisfy the following mechanical stability restrictions [23]: Thus, we can assert that the hexagonal MAuGe (M = Lu, Sc) is in a mechanically stable state for pressure range 0-40 GPa. The fit results are given by the following expressions for both compounds LuAuGe and ScAuGe, respectively:     (3.13)

Elastic constants for polycrystalline aggregates
The isotropic elastic parameters can fully describe the mechanical behaviour of a polycrystalline material using one of the three pairs of isotropic elastic parameters: either the bulk modulus with the shear modulus , the two Lamé's constants and or the Young's modulus with the Poisson's ratio . Theoretically, the two isotropic elastic parameters and of the polycrystalline phase of a material can be obtained by a special averaging of the individual elastic constants of the monocrystalline phase. The Reuss-Voigt-Hill approximations [25,26] are the most used. Voigt ( V , V ) and Reuss ( R , R ) approximations represent the extreme values of and for polycrystalline samples. The two isotropic elastic parameters and are expressed for hexagonal systems as follows [23]: (3.14) Hill recommends that the arithmetic mean of the Voigt and Reuss limits should be used in practice as an efficient model for determining the isotropic elastic parameters of polycrystalline samples: where H and H are the bulk and shear moduli, respectively, of the polycrystalline material according to Hill's approximation. The Young's modulus and the Poisson's ratio for anisotropic material can be calculated from H and H using the following expressions:    table 7 for both LuAuGe and ScAuGe is greater than 1.75, which suggests that both MAuGe are ductile. Thus, they will be resistant to thermal shocks. (iv) The Poisson's ratio is generally related to the volume change in a solid during uniaxial strain [28][29][30]. From the values of in table 7, the smallest calculated values are 0.28 for LuAuGe and 0.26 for ScAuGe, which shows that a considerable change in volume can be associated with elastic deformation in the considered materials.
For a complete description of the mechanical properties of MAuGe ( M = Lu, Sc), we also computed the isotropic longitudinal , transverse and average sound wave velocities using the following relations [31,32]: where, is the bulk modulus, is the shear modulus and is the mass density. We also estimated the Debye temperature D , which is an important physical parameter. The Debye temperature D is defined in terms of the average sound velocity as follows [29,31]: where, ℎ and B are the Planck constant and Boltzmann constant, respectively, A is the Avogadro number, is the mass density, is the molecular weight and is the number of atoms per unit cell.

Elastic anisotropy
Elastic anisotropiy has an important implication in the engineering science. Recent research shows that the elastic anisotropy for solid crystals has an influence on microcracks in materials [29,33] and on the nanoscale precursor textures in alloys [34,35]. Different criteria have been developed to describe the elastic anisotropy of materials. (i) For a hexagonal structure, the anisotropic shear factors 1 , 2 , and 3 provide a measure of the degree of anisotropy for the bonding between atoms in different planes. For an isotropic crystal, 1 , 2 and 3 should have values equal to unity, while any value other than unity is an indication of elastic anisotropy. The elastic anisotropy factors 1 , 2 and 3 can be expressed as follows: (ii) Another way to evaluate the elastic anisotropy consists in introducing the Voigt and Reuss bounds [33]. The elastic anisotropy in compression (shear) defined by the factor ( ) is expressed as follows: (3.22) where and are the bulk and shear moduli, and the subscripts V and R represent the Voigt and Reuss bounds. The and ratios can range from zero to 100%. A value of zero represents elastic isotropy and a value of 100% represents the largest possible elastic anisotropy.
(iii) The third way consists in precise quantifying the extent of the elastic anisotropy using the universal index [36]. The index takes into account both compression and shear contributions, which is defined as follows: (3.23) The universal index is equal to zero ( = 0) for isotropic crystals, and the deviation of from zero shows the presence of elastic anisotropy.
The elastic anisotropy values deduced from the factors 1 , 2 , 3 , , and are given in table 9. From this table, we see that 3   anisotropy is relatively more pronounced than the shear anisotropy for both compounds. The values also indicate the presence of elastic anisotropy for the two studied materials.

Electronic properties
The Brillouin zone (BZ) which highlights the selected path Γ-A-H-K-Γ-M-L-H to calculate the energy band structures for the MAuGe (M=Lu, Sc) compounds, is illustrated in figure 10. The MAuGe band structures along the chosen path are illustrated in figure 11. It is seen that the LuAuGe and ScAuGe compounds at their equilibrium lattice parameters have similar energy band dispersions in the considered energy range (from −12 eV to 4 eV) with some small differences depending on the electron valence states of the Lu/Sc,Au and Ge atoms. It should be noted that the valence and conduction bands overlap at the Fermi level (EF), which reveals the absence of the bandgap at the Fermi level. Consequently, the MAuGe (M = Lu, Sc) compounds exhibit a metallic nature.
In order to determine the contribution of the electron valence states of each atom in the MAuGe electronic band structures, we calculated the total (TDOS) and partial (PDOS) densities of states for both compounds. The TDOS and PDOS diagrams are shown in figure 12. It is clear that the total density of states for MAuGe is characterized by two distinct regions in the −12 eV to 0 eV energy range. The first region is located between −12 and −9 eV and is mostly derived from the Ge-s states for both studied compounds. The second region starts from about −7.5 eV up to the Fermi level ( ) and is mainly composed of the Lu-f and Au-d and Ge-p states. The lowest conduction bands (from 0 to 4 eV) are mostly made up of the Lu/Sc-d unoccupied states.
To better understand the chemical bonding character between the Lu/Sc, Au and Ge atoms, the electron density distribution maps in the crystallographic plane (110) for MAuGe (M = Lu, Sc) compounds are shown in figure 13 together with the calculated bond lengths that are listed in table 4. One can see that the Ge1-Au2 and Ge1-Au1 bonds are characterized by an obvious deformation of the electron charge density distribution (see the yellow area in the electron density maps between Ge and Au atoms), which indicates that the Ge1-Au2 and Ge1-Au1 bonds have a covalent bonding nature for both compounds studied. The hybridization of the Ge-p and Au-p states, which is clearly visible in the PDOS spectra
To further explore the MAuGe (M = Lu, Sc) electronic structures, we calculated the atomic charges of M, Au and Ge atoms using Hirshfeld's population analysis [37]. The obtained results are tabulated in table 10. One can observe that M = Lu/Sc, Au and Ge atoms have small effective charges (positive charges for M and Au, and negative charge for Ge). A very lower atomic charge difference implies much lower ionicity and higher covalency in the corresponding chemical bonds. Thus, the chemical bonding between Ge and Au is covalent.

Conclusions
In summary, we have performed ab initio calculations of the structural, electronic and elastic properties for the MAuGe (M = Lu, Sc) compounds by means of the pseudopotential plane-wave method in the framework of the density functional theory within the generalized gradient approximation. Our results can be summarized as follows: ⊲ The optimized structural parameters are in very good agreement with the existing experimental and theoretical data.
⊲ The elastic constants of the monocrystalline phase calculated at zero pressure show that the MAuGe (M = Lu, Sc) materials are mechanically stable. Note that the mechanical stability remains verified for hydrostatic pressures up to 40 GPa.
⊲ The numerical estimates of the elastic moduli of the polycrystalline phase, i.e., Young's modulus, shear modulus, Poisson's ratio, anisotropy factors, sound velocities and Debye temperature were evaluated and discussed under pressure for the first time. The Pugh's ratio for LuAuGe and ScAuGe indicates that these materials are ductile.