The body centered tetragonal LaRu₂As₂ structure crystallizes with space group I4/mmm (No. 139) as shown in Fig. 1. Each atomic species in this crystal occupies only one crystallographic position and only one internal parameter z_X settles the relative position of As inside the unit cell. There are two formula units in each unit cell. The 2a, 4d, and 4e atomic positions are inhabited by the La, Ru, and As atoms, respectively. The negatively charged [Ru₂As₂] blocks and positively charged La layers take positions by turns along the c-direction and form the crystal structure. The optimized lattice constants and z_X parameter of LaRu₂As₂ are given in Table 1 along with those obtained for some other 122 phase compounds. The computed lattice parameters accord well with the measured values.^[11] Moreover, the calculated structural parameters are slightly larger (a ∼ 0.23%, c ∼ 2.94%, and V ∼ 3.41%) than the experimental values, which is a general trend inherent to GGA calculations.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) Crystal structure of new 122 phase superconductor LaRu2As2.

Table 1.

Table 1.

Table 1. Structural properties of LaRu2As2 in comparison with those for the related ThCr2Si2-type compounds. . Compounds a/Å c/Å V/Å3 c/a zX References LaRu2As2 4.1923 10.9014 191.59 2.6003 0.3641 This 4.1826 10.5903 185.27 2.5320 – [11] BaRu2As2 4.1925 12.3136 216.44 2.9371 0.3510 [22] 4.1525 12.2504 211.24 2.9501 0.3527 [23] SrRu2As2 4.2068 11.2903 199.81 2.6838 0.3591 [22] 4.1713 11.1845 194.61 2.6813 0.3612 [23]

Table 1. Structural properties of LaRu2As2 in comparison with those for the related ThCr2Si2-type compounds. .

The mechanical behavior of solids can be described successfully by the elastic constants that are related to the material’s response to an applied stress. The finite strain theory assigned in CASTEP code is well established for calculating the elastic constants successfully for a range of materials including metallic systems.^[24] In this method, a given uniform deformation ε_j is used and then the resulting stress σ_i is calculated. By choosing an appropriate functional deformation, elastic constants are then evaluated by solving the linear equation, σ_i = C_ijε_j.

The well-established Voight–Reuss–Hill (VRH) approximations validated in many metallic and insulating materials^[25–27] are used to calculate the polycrystalline bulk elastic properties, namely, bulk modulus B and shear modulus G from calculated C_ij. Further, the equations Y = (9GB) / (3B + G) and v = (3B − 2G)/(6B + 2G) are employed to determine the Young’s modulus Y and Poisson’s ratio v, respectively.

The values obtained for single crystal elastic constants (C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, and C₆₆) and bulk elastic properties (B, G, Y, B/G, and v) of tetragonal crystal system LaRu₂As₂ are listed in Table 2 along with the results found in the literature for La-based isostructural compounds.^[28] The mechanical stability conditions for tetragonal crystals^[29] are as follows: C₁₁ > 0, C₃₃ > 0, C₄₄ > 0, C₆₆ > 0, C₁₁ − C₁₂ > 0, C₁₁ + C₃₃ − 2C₁₃ > 0, and 2(C₁₁ + C₁₂) + C₃₃ + 4C₁₃ > 0. These criteria are valid for the newly synthesized 122 phase, indicating the mechanical stability of the compound. The elastic constants C₁₁ and C₃₃ explain the linear compression resistances along the a- and c-directions, respectively. It is evident that the obtained elastic constants C₁₁ and C₃₃ are very large in comparison to other elastic constants, implying that the LaRu₂As₂ crystal should be very incompressible under uniaxial stress along both a- and c-directions. Again, the elastic constant C₁₁ is much larger than C₃₃, meaning that the incompressibility along the a-direction should be much higher than that along the c-direction. In fact, the bonds aligned to the a-axis contribute a dominating effect on C₁₁, making it much larger than C₃₃. Since C₁₁ + C₁₂ > C₃₃, the bonding in the (001) plane should exhibit more elastic rigidity than that along the c-axis, and the elastic tensile modulus should be higher on the (001) plane than that along the c-axis. The elastic constant C₄₄ indicates the ability to resist the shear distortion in (100) plane, whereas the elastic constant C₆₆ is correlated with the resistance to shear in the 〈110〉 direction.^[30] Since C₆₆ > C₄₄, the new compound should be more competent to resist the shear distortion in the 〈110〉 direction than in the (100) plane. The shear anisotropy factor A, defined as A = 2C₆₆/(C₁₁ −C₁₂) is 1.24, signifying that the shear elastic properties of the (001) plane in LaRu₂As₂ depend on the shear directions significantly.

Table 2.

Table 2.

Table 2. Single crystal elastic constants Cij, polycrystalline bulk modulus B, shear modulus G, and Young’s modulus Y, in GPa, Pugh’s ratio G/B, and Poisson’s ratio v of LaRu2As2. . Compounds C11 C12 C13 C33 C44 C66 B G Y B/G v A Ref. LaRu2As2 244 86 82 113 62 98 113 6 159 1.79 0.265 1.24 This LaNi2Ge2 152 87 40 126 87 90 62 74 159 0.84 0.073 2.77 [28] LaNi2P2 202 116 15 102 116 97 62 93 186 0.67 0.002 2.26 [28]

Table 2. Single crystal elastic constants Cij, polycrystalline bulk modulus B, shear modulus G, and Young’s modulus Y, in GPa, Pugh’s ratio G/B, and Poisson’s ratio v of LaRu2As2. .

The concept of failure mode, i.e., ductile or brittle nature, of materials is important in the study of mechanical behaviors of solids. A material can be either brittle or ductile in most practical situations. To quantify the failure state of solids, Cauchy pressure, and Pugh and Poisson’s ratios are used as powerful tools. A brittle material changes its volume easily and a ductile material can be easily distorted under the action of external loads. The Cauchy pressure^[31] defined as (C₁₂−C₄₄) identifies a solid as either a brittle or ductile one. When Cauchy pressure is positive (negative), the material is prone to being ductile (brittle). In accordance with Pugh’s ratio,^[32] the high (low) B/G value signifies the ductile (brittle) nature of the materials. The threshold value of B/G is found to be 1.75, which separates the ductile materials from brittle ones. Frantsevich et al.^[33] separated the ductile materials from brittle materials in terms of Poisson’s ratio. This rule proposes v ∼ 0.26 as the border line that separates the brittle and ductile materials. A material having the Poisson’s ratio greater than 0.26 will be ductile, otherwise the material will be brittle.

The Cauchy pressure (C₁₂−C₄₄) of LaRu₂As₂ is positive, its Pugh’s ratio B/G is 1.79 which is greater than 1.75 and its Poisson’s ratio v is 0.265 which is greater than 0.26. As a result, the compound LaRu₂As₂ should behave as a ductile material. It should be mentioned that there are some papers^[34–37] in which B/G = 2.0 (G/B = 0.5) and v ∼ 0.33 are used as critical values for separating the brittle materials from ductile ones. On the other hand, many authors^[38–44] used the values presented in this paper to judge the brittle/ductile nature of the materials. In fact, Vaitheeswaran et al.^[39] preferred B/G = 1.75 as threshold value for defining the brittle/ductile nature of the materials and showed that Pugh’s critical value corresponds to v = 0.26 as the border line between the ductile and brittle materials. These criteria are frequently used to identify the brittle material from ductile ones.

For describing the elastic behaviors of solids, bulk and shear moduli are two important parameters. Fracture and plastic deformation are two essential issues related to solid materials. Bulk modulus B measures the resistance to fracture and shear modulus G evaluates the resistance to plastic deformation of polycrystalline materials.^[45] From dislocation theory^[46] it is known that a weak relationship is observed between bulk modulus and hardness. On the contrary, a good relationship exists between hardness and shear modulus.^[47] Certainly, the hardness is less susceptible to the bulk modulus than the shear modulus. Therefore, as can be seen from Table 2, the hardness of LaRu₂As₂ is expected to be small and LaRu₂As₂ should be soft and easily machinable.

Many important issues related to physical properties of solids can be addressed with Young’s modulus. The resistance in opposition to longitudinal tension can be estimated with Young’s modulus Y. The Young’s modulus also manipulates the thermal shock resistance of solids because of having the relation between critical thermal shock coefficient R and Y. In fact, the critical thermal shock coefficient R is found to be changed in inverse proportion to the Young’s modulus Y.^[48] The larger the R value, the better the thermal shock resistance. The thermal shock resistance is a fundamental parameter for thermal barrier coating (TBC) material selection. In comparison with a promising thermal barrier coating material Y₄Al₂O₉,^[49] the small Y value of LaRu₂As₂ signifies that it should have a reasonable resistance to thermal shock.

Debye temperature θ_D is a distinctive temperature of crystals at which the highest-frequency mode (and hence all modes) of vibration is excited. It controls many physical properties of materials and is used to make a division between high-and low-temperature regions for a solid. To distinguish the classical and quantum behavior of phonons, the Debye temperature also defines a border line between them. When the temperature T of a solid is elevated over θ_D, each mode of vibrations is expected to be associated with an energy equal to k_BT. At T < θ_D, all high-frequency modes seem to be in sleep. The vibrational excitations at low temperature are due to acoustic vibrations. The Debye temperature, θ_D, is determined by using one of the standard methods depending on the elastic moduli.^[50] This method guides us in calculating the Debye temperature from the average sound velocity vm by the following equation: θ_D = h/k_B[(3n/4π)N_Aρ/M]^1/3v_m, where h refers to the Planck’s constant, k_B represents the Boltzmann’s constant, n is the number of atoms per formula unit, M is symbolized for molar mass, N_A is the Avogadro’s number, and ρ denotes the mass density of solid. The average sound velocity within a material is obtained by . Here vl and vt indicate the longitudinal and transverse sound velocities in crystalline solids and are expressed as v₁ = [(3B + 4G)/3ρ]^1/2 and v_t = [G/ρ]^1/2, respectively.

The Debye temperature in cooperation with longitudinal, transverse and average sound velocities calculated within the present formalism is listed in Table 3. The values of v_l, v_t, v_m, and θ_D found in Ref. [28] are calculated by using elastic moduli B and G within the Voigt approximation. Here we convert these values into aggregate values using Hill approximations.^[27] As a rule of thumb, a lower Debye temperature involves a lower phonon thermal conductivity. The examined phase LaRu₂As₂ has the lower Debye temperature than the other two isostructural compounds as well as a candidate material for thermal barrier coating Y₄Al₂O₉^[49] and hence has a lower thermal conductivity. Accordingly, LaRu₂As₂ should have opportunity to be used as a thermal barrier coating (TBC) material.

Table 3.

Table 3.

Table 3. Calculated values of density (ρ in gm/cm3), longitudinal, transverse, and average sound velocities (vl, vt, and vm in km/s) and Debye temperature (θD in K) of LaRu2As2. . Compounds ρ vl vt vm θD Refs. LaRu2As2 8.799 4.731 2.676 2.976 335 This LaNi2Ge2 7.685 4.572 3.103 3.384 389 [28] LaNi2P2 6.853 5.210 3.684 3.994 478 [28]

Table 3. Calculated values of density (ρ in gm/cm3), longitudinal, transverse, and average sound velocities (vl, vt, and vm in km/s) and Debye temperature (θD in K) of LaRu2As2. .

Understanding of electronic features (band structure, DOS, etc) is important to explain many physical phenomena, namely optical spectra and transport properties of solids. The full picture of energy bands and band gaps of a solid is known as electronic band structure or simply band structure. Actually, in solid-state and condensed matter physics, the band structure defines certain ranges of energy, which are allowed for electrons within a solid, and the ranges of energy, which are not allowed for any electrons. The number of bands indicates the available number of atomic orbitals in the unit cell. The calculated energy band structure with high symmetry points in the first Brillouin zone using equilibrium lattice parameters is shown in Fig. 2(a). The dashed line in the band structure corresponds to the energy of the highest filled state. This energy level is known as Fermi level, E_F. Depending on the position of E_F, several important distinctions regarding the expected electrical conductivity of a material can be made. The Fermi level of the new 122 phase superconductor takes position just above the valence band maximum near the Γ point. The new compound should have metallic behaviors in nature since its valence bands cross the Fermi level and overlap noticeably with the conduction bands. In addition, no band gap is found in the Fermi level. The near-Fermi bands show a complicated ‘mixed’ character, combining the quasi-flat bands with a series of high-dispersive bands intersecting the Fermi level. The detailed features of band structure can be explained with the calculated total and partial densities of states.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) Electronic structures of LaRu2As2, showing (a) electronic band structure, (b) total and partial densities of states.

The calculated total and partial densities of states are shown in Fig. 2(b), where the vertical broken line denotes the Fermi level, E_F. At the left of the Fermi level, a deep valley is observed, which is known as pseudogap E_p. The delocalized electrons occupy the levels above E_p and the materials containing such electrons turn into metalized ones. It is seen that the Fermi level of LaRu₂As₂ lies in a region of relatively low DOS. Consequently, the new compound should be stable in perspective of electronic features as stated in the free electron model.^[51] The new 122 phase superconductor LaRu₂As₂ should exhibit metallic conductivity due to having finite value of total density of states (TDOS) at E_F. The atom resolved partial DOS gives an insight into the electronic structure. Mainly Ru 4d states contribute to the DOS at the Fermi level along with little contribution from La 5d and As 2p states. At E_F the calculated TDOS has a value of 5.4 states per unit cell per eV. The large TDOS is a sign of high metallicity of the compound. The TDOS at the Fermi level is quite high compared with those of some other recently discovered ternary superconducting compounds.^[52,53] This may be responsible for the higher superconducting transition temperature in LaRu₂As₂, than, for example, in ternary silicides.^[54]

The lowest lying valence bands extended from −17.9 to −16.3 eV arise entirely from La 5p states. These valence bands are observed to be separated by a wide prohibited energy gap of ∼ 3.3 eV from the next valence bands lying in an energy range from −13.0 eV to −10.6 eV. These valence bands are again separated by a forbidden energy gap of width 3.4 eV from the highest valence bands. The highest valence bands contain three visible peaks and broaden in an energy range from −6.6 eV to Fermi level. The left peak originates mainly from As 4p and Ru 4d states. The middle and highest peak are due to equal contribution from Ru 4d and As 4p states. The first peak near to Fermi level forms due to a contribution coming from Ru 4d states. In the energy range between −6.6 eV and the Fermi level, the covalent bonding between comprising elements is expected. This is due to the reason why the states are truly degenerate regarding angular momentum and lattice site. Because of the difference in the value of electronegativity among the constituent elements, few ionic interactions can be predicted within LaRu₂As₂ as well.

The allocation of electrons in several fractional manners among the various parts of bonds can be explained by analyzing the Mulliken population. The overlap population provides a relation between covalency/ionicity of bonding and bond strength. Sanchez-Portal et al.^[55] developed a method of analyzing the population in CASTEP code using a projection of the plane wave states onto a localized basis. The Mulliken scheme^[56] is then applied to the population analysis of the resulting projected states. Analyzing the electronic structure calculations with linear combination of atomic orbital (LCAO) basis sets, this technique is used extensively. Using minimal basis Mulliken scheme^[56,57] provides two fundamental quantities regarding atomic bond population, i.e., the effective charge and the bond order values between a pair of bonding atoms, which are expressed as

To visualize the nature of chemical bonding in LaRu₂As₂, the electron charge density distribution is calculated and the contour of electron charge density is presented in Fig. 3. On an adjacent scale, the blue and red colors signify the low and high electron densities, respectively. An atom with large electronegativity (electronic charge) exerts a pull on electron density towards itself.^[63] Because of large difference in electronegativity and radius of atoms, the electronic charges around Ru (2.20) and As (2.18) are greater than that around La (1.10). The higher electronegativities of Ru and As exhibit strong accumulation of electronic charge while relatively low charge density indicates weak charge accumulation of La. A typical ionic picture based on standard oxidation states of atoms can help to explain the bonding nature of a compound. In LaRu₂As₂, the oxidation states of atoms are La²⁺, Ru²⁺, and As³⁻. The sum of the oxidation numbers must be zero for a neutral compound such as LaRu₂As₂. In the LaRu₂As₂ structure, the [Ru₂As₂] blocks are separated by La atomic sheets. The charge states of these sheets/blocks are [La]²⁺ and [(Ru²⁺)₂(As³⁻)2]²⁻. It can be said that a charge transfer will take place from [La]²⁺ sheets to [Ru₂As₂]²⁻ blocks. The existence of ionic bonds between Ru–As within [Ru₂As₂]²⁻ blocks can be guessed from the balance between negative and positive charges at the positions of relevant atoms. From atom resolved DOS it is observed that the hybridization between Ru 4d and As 4p states leads to the formation of covalent Ru–As bonds. Moreover, metallic-type Ru–Ru bonds are expected to exist in [Ru₂As₂] blocks due to overlapping of Ru 4d states near Fermi level (Fig. 2(b)). Surprisingly, no As–As bonds exist between two adjacent [Ru₂As₂] blocks in LaRu₂As₂. But such bonds are found to be present in (Ca/Ba)Fe₂As₂ iron pnictides.^[64] The possible reason for this contradiction may be explained as follows: the low strength of Fe–As bond in iron pnictide in comparison with Ru–As bonds in LaRu₂As₂ favors the formation of As–As bond in (Ca/Ba)Fe₂As₂. The above discussion reflects that the chemical bonding in LaRu₂As₂ can be described as a highly anisotropic combination of ionic, covalent and metallic interactions.

The Fermi surface topology is calculated, which is presented in Fig. 3(b). As is observed, the near-Fermi surface band picture has a complicated ‘mixed’ character: the quasi-flat band and the high-dispersive band intersect the Fermi level simultaneously. These features concede a multi-sheet Fermi surface consisting of two quasi-two-dimensional electron-like sheets in the corners of the Brillouin zone. The nearest sheet is also parallel to A–M direction and the distant sheet is half-folded and intersected at the centre of the sheet by the Γ – M line. Between these two sheets a concave sheet appears and the Γ –M line serves as the axis of this three-dimensional sheet. The central electron-like sheet is very complicated and has four half-tube shaped wings cutting along its own axis. These four wings form a red-cross shape keeping each arm along the Z–R direction. The existence of the multiband nature suggests that the new compound LaRu₂As₂ could be a class of multiple-gap superconductor.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) Charge density (a) and Fermi surface (b) of LaRu2As2.

Accurate value of electron–phonon coupling constant (λ) is important for evaluating the superconducting transition temperature T_c. The method assigned to QUANTUM ESPRESSO program^[65] can accurately compute the λ value directly. We stress that a double-delta function integration over a dense net of electron and phonon vectors (k and q) is required to compute the electron–phonon coupling constant with significant computational resources. Moreover, this method is more complicated for a crystal system containing many atoms in its unit cell like LaRu₂As₂. All of these prohibit us from carrying on with this method. We have an alternative method in which the equation provides the electron–phonon coupling constant.^[66] However, in the present situation we have no experimental values for electronic specific-heat coefficient γ and theoretically calculated value yields a lower value of λ.^[67] So, we have only one indirect method based on McMillan’s equation to calculate the value of λ as follows:

The repulsive Coulomb potential µ^∗ is assigned to a value in a range of 0.10–0.15 and the choice of µ^* is fairly arbitrary. Using known T_c = 7.8 K^[11] with our calculated Debye temperature θ_D = 335 K in McMillan’s equation, we have λ = 0.64 and 0.76 for µ^* = 0.1 and 0.15, respectively, which implies that LaRu₂As₂ should be a moderately coupled BCS superconductor.