† Corresponding author. E-mail:
In this paper, we perform the density functional theory (DFT)-based calculations by the first-principles pseudopotential method to investigate the physical properties of the newly discovered superconductor LaRu2As2 for the first time. The optimized structural parameters are in good agreement with the experimental results. The calculated independent elastic constants ensure the mechanical stability of the compound. The calculated Cauchy pressure, Pugh’s ratio as well as Poisson’s ratio indicate that LaRu2As2 should behave as a ductile material. Due to low Debye temperature, LaRu2As2 may be used as a thermal barrier coating (TBC) material. The new compound should exhibit metallic nature as its valence bands overlap considerably with the conduction bands. LaRu2As2 is expected to be a soft material and easily machinable because of its low hardness value of 6.8 GPa. The multi-band nature is observed in the calculated Fermi surface. A highly anisotropic combination of ionic, covalent and metallic interactions is expected to be in accordance with charge density calculation.
A great variety of compounds with RT2X2 stoichiometry possess the body-centered tetragonal ThCr2Si2-type structure (space group I4/mmm, No. 139).[1] In this stoichiometry, R is symbolized for a rare earth, alkaline earth or alkali element, T refers to a transition metal, and X represents p-metal atoms, namely B, P, Si, As, or Ge. This large family includes more than 700 members of 122-type intermetallics (so-called 122 phases) and exhibits an outstanding collection of physical and chemical properties.[2] The structure of RT2X2 stoichiometry offers an innate multilayer system in which the planes of R ions are taken apart from T metallic layers by X atomic sheets, comprising R–X–T–X–R stacking of individual basal planes along the c axis. The ThCr2Si2-type structure exhibits a strong uniaxial anisotropy with keeping c axis as an anisotropy axis. In other words, the layer of edge sharing TX4 tetrahedron that is parallel to the ab plane and separated by the basal plane of R atoms characterizes the structure of RT2X2. In this structure, the R atomic sheets separate into two adjacent [T2X2] blocks. In sequence, T ions inside [T2X2] blocks build up a square lattice pressed between two X sheets shifted in order that each T is enclosed by a distorted X tetrahedron TX4. A wide range of variation occurs in the interlayer distance between X atoms when either R or T is changed for a particular p-metal atom.[2,3]
This group of materials has attracted a lot of attention in recent years because of the discovery of high-temperature superconductivity in iron arsenide AFe2As2 (A = K, Ca, Sr, Ba, etc.).[4,5] As Ru lies in the same group together with Fe in the periodic table, the superconductivity in the level of iron arsenide is anticipated for Ru-related compounds, and several studies have been devoted to these materials.[6–10] Motivated by this, Guo et al.[11] have recently synthesized the polycrystalline samples of ThCr2Si2-type LaRu2As2 and reported the bulk superconductivity in LaRu2As2. Their temperature-dependent resistivity measurement ensures a superconducting transition of LaRu2As2 with an onset Tc of 7.8 K. This measurement also estimates the upper critical field µ0Hc2(0) at zero temperature and obtains a value of 1.6 T. Further, direct current (DC) magnetic susceptibility measurement detects a bulk superconducting Meissner transition at 7.0 K, and the isothermal magnetization measurement signifies that LaRu2As2 is a type-II superconductor. Surprisingly, LaRu2As2 has the highest transition temperature Tc among the iron-free transition metal pnictides with the ThCr2Si2-type crystal structure.
In this paper, the study of LaRu2As2 is carried out via the ground state electronic structure calculations by using the plane-wave pseudopotential approach within the density functional formalism. The aim of the present paper is to investigate the ground state properties including structural, elastic, electronic, and bonding characters of the newly discovered 122 phase compound by calculating its total energy in an optimized volume. To the best of our knowledge, this is the first ab initio calculation completed on this new superconductor LaRu2As2.
The rest of this paper is divided into three sections. In Section
The approach used to investigate the physical properties of LaRu2As2 as implemented in the Cambridge Serial Total Energy Package (CASTEP) code[12] is as follows: a set of Kohn–Sham equations are solved by means of the plane-wave pseudopotential method[13] within the strong electron ensemble density functional theory (DFT).[14] Exchange and correlation energies are described by a nonlocal correction for local density approximation (LDA) in the form of generalized gradient approximation (GGA).[15] The wave-functions are allowed to expand in a plane wave basis set by following the periodic boundary conditions and Bloch’s theorem.[16] The electron–ion potential is treated by means of first-principles pseudopotentials within Vanderbilt-type ultrasoft formulation.[17] Broyden–Fletcher–Goldfarb–Shanno (BFGS) energy minimization technique[18] best for crystalline materials is used to find, self-consistently, the electronic wave-functions and consequent charge density. In addition, the density mixing[19,20] scheme is employed. For k-points sampling integration over the first Brillouin, the Monkhorst–Pack scheme[21] is used. The parameters that are chosen for the present calculations are as follows: energy cutoff is 500 eV (340 eV for elastic property calculations), k-point mesh is 16 × 16 × 7, the difference in total energy per atom is within 5 × 10−6 eV, maximum ionic Hellmann–Feynman force is within 0.01 eV/Å, maximum stress is within 0.02 GPa, and maximum ionic displacement is within 5 × 10−4 Å.
In this section the investigated physical properties of the 122 phase superconductor LaRu2As2 are presented and analyzed; among them are the structural and elastic properties, Debye temperature, band structure and density of states (DOS), and then Mulliken population and Vickers hardness and the electron charge density as well as Fermi surface of LaRu2As2 under zero pressure.
The body centered tetragonal LaRu2As2 structure crystallizes with space group I4/mmm (No. 139) as shown in Fig.
![]() | 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 εj is used and then the resulting stress σi is calculated. By choosing an appropriate functional deformation, elastic constants are then evaluated by solving the linear equation, σi = Cijεj.
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 Cij. 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
![]() | 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
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 θD 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 θD, each mode of vibrations is expected to be associated with an energy equal to kBT. At T < θD, all high-frequency modes seem to be in sleep. The vibrational excitations at low temperature are due to acoustic vibrations. The Debye temperature, θD, 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: θD = h/kB[(3n/4π)NAρ/M]1/3vm, where h refers to the Planck’s constant, kB represents the Boltzmann’s constant, n is the number of atoms per formula unit, M is symbolized for molar mass, NA is the Avogadro’s number, and ρ denotes the mass density of solid. The average sound velocity within a material is obtained by
The Debye temperature in cooperation with longitudinal, transverse and average sound velocities calculated within the present formalism is listed in Table
![]() | 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.
![]() | 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.
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
![](cpb_26_3_037103/cpb_26_3_037103eqn13.gif)
![](cpb_26_3_037103/cpb_26_3_037103eqn14.gif)
![](cpb_26_3_037103/cpb_26_3_037103eqn15.gif)
![](cpb_26_3_037103/cpb_26_3_037103eqn16.gif)
![](cpb_26_3_037103/cpb_26_3_037103eqn17.gif)
![](cpb_26_3_037103/cpb_26_3_037103eqn18.gif)
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.
The Fermi surface topology is calculated, which is presented in Fig.
Accurate value of electron–phonon coupling constant (λ) is important for evaluating the superconducting transition temperature Tc. 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
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 Tc = 7.8 K[11] with our calculated Debye temperature θD = 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.
DFT method is used within the first-principles pseudopotential method to calculate some physical properties of newly discovered superconductor LaRu2As2. The evaluated lattice constants show sound agreement with the experimental results. The obtained single crystal elastic constants obey the mechanical stability conditions for the new tetragonal system. The LaRu2As2 crystal shows high uniaxial elastic anisotropy. The newly synthesized compound is expected to resist the shear distortion in the 〈110〉 direction more strongly than in the (100) plane. Cauchy pressure, Poisson’s ratio, and Pugh’s ratio indicate that LaRu2As2 should be a ductile material. The relatively small Young’s modulus indicates that LaRu2As2 should attain reasonable resistance to thermal shock. The low Debye temperature also suggests that this compound should be favorable for use as a thermal barrier coating material. The calculated electronic structures reveal that the intra-atomic bonding in new 122 phase compound LaRu2As2 may be explained as a mixture of ionic, covalent and metallic interactions. The hardness value of 6.8 GPa implies that this material should be soft and easily machinable. The obtained Fermi surface exhibits the multi-band nature, suggesting that the new ternary compound LaRu2As2 could be a kind of multiple-gap superconductor. The calculated λ indicates the strong electron–phonon coupling in LaRu2As2. Very recently, we have noted that some ab initio calculations have been done on LaRu2As2 and LaRu2P2 compounds.[68] Some of the basic features of those calculations agree quite well with the ones presented in this paper. To conclude, we expect that this study will stimulate further experimental and theoretical research activities on the newly synthesized LaRu2As2 compound.
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