Insight into the mechanical, thermodynamics and superconductor properties of NbRuB via first-principles calculation

Using the first-principles calculations, the electronic structure, chemical bonding, mechanical, thermodynamics and superconductor properties of NbRuB are investigated. The optimized lattice parameters were in good agreement with the experimental data. The analysis of the density of states and chemical bonding implies that the metallic behavior of NbRuB originates from the Ru and Nb, and the bonding behaviors are a mixture of covalent-ionic bonds. The bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio and hardness of NbRuB were calculated. The results reveal that the NbRuB is ductility and the Vickers hardness is 15.06 GPa. Moreover, the 3D dependences of reciprocals of Young’s modulus is also calculated and discussed, showing strong anisotropic character for NbRuB. Finally, the Debye temperature and superconducting transition temperature are obtained.


Results and Discussions
Electronic Structure and Chemical Bonding. NbRuB has an orthorhombic structure with space group Pmma, as shown in Fig. 1. It is characterized by two different layers alternately stacked: The first layer contains Nb and B atoms, whereas the second layer is filled with Ru atoms and the B2 dumbbells. In NbRuB, there are two types of B atoms: one is in the center of a triangular prism with Ru at the six vertices, and the other exists as a B-B dimer inside a double-triangular prism of Nb. The optimized lattice parameters a = 10.833 Å, b = 3.141 Å and c = 6.324 Å are in well agreement with the experimental data reported in ref. 4 and ref. 7 (Table 1). There is a slightly underestimate of only 0.5% from the experimental data, which is due to the thermal expansion effects.
To gain further insights into NbRuB, the band structure, total density of states (TDOS) and partial density of states (PDOS) of Nb, Ru and B are plotted in Fig. 2. From − 8 eV to the E f , the majority of the DOS stems from 4d states of Ru and Nb, followed by B-2p states. The conduction bands above the E f originate mostly from Nb-4d and B-2p states, and small contributions from s and p states of Ru, Nb and B. From Fig. 2, the NbRuB exhibit metallic behavior because there is no band gap at the E f . Though electronic structure analysis, the metallic behavior of NbRuB originates from Nb and Ru metals which contribute nearly equally at the E f of the DOS. The metallic behavior of NbRuB indicates that this material might be a superconductor and we will discuss it in the following.
Population analysis provides more insightful information on the chemical bonding characteristics. A high value of the bond population indicates a covalent bond, while a low value indicates an ionic interaction. In this work, the Mulliken method is used to calculate the overlap population and the charge, and the calculated results and the experiment data are listed in Table 2 Mulliken's bond overlap population for nearest neighboring atoms is a measure of spatial charge density between bonding atoms, and this quantity does not totally depend on the valence charge transfer along the bond  axis. The bond overlap populations can reflect the trend of bond ionicity. The ionicity of a bond based on bond overlap population can be calculated as follows 11 : Where P is the overlap population of bond, P c is the overlap population of the bond in a pure covalent crystal (We assume the P c = 1 for purely covalent bond). The f h = 0 for a purely covalent bond, while f h = 1 indicates a purely ionic bond. The bond length, population and population ionicity of the NbRuB are given in Table 2. From Table 2, it is clear that the strongest chemical interaction in NbRuB is the B1-B1 dumbbell, and the B-Ru shows a high level of covalency and a low level of ionicity, while the Ru-Ru, Nb-Ru and B-Nb shows almost a complete ionicity ( f h ≈ 1). Thus, we conclude that the bonding behaviors of NbRuB may be described as a mixture of covalent-ionic bonds.
Mechanical properties. The elastic constants not only provide a link between the mechanical and dynamical behavior of crystals, but also provide important information concerning the nature of the forces operating on solids. The elastic constants of NbRuB by the strain-stress method are listed in Table 3. Since the elastic constants C 11 , C 22 and C 33 can be directly related to the crystallographic x, y, and z axes, respectively. As seen in Table 3, the observed ordering of elastic constants is C 11 ≈ C 33 > C 22 , which represents a relative weakness of lattice interactions along the crystallographic y axis. Moreover, The Young's modulus for the different directions have also been determined ( = / E s 1 ii ii , i = 1, 2 or 3, where s ii are the diagonal elements of the compliance matrix). The calculated    (100), (010) and (001) planes. In Table 3, C 55 is smaller than C 44 and C 66 , indicating the soft shear transformation along the (010) plane.
To be mechanically stable, the elastic constant should satisfy the generalized elastic stability criteria. For the NbRuB crystals, the stability criteria are given as follows 12 : Clearly, the calculation results suggest that NbRuB is mechanically stable at the ambient condition. According to the calculated elastic constants, the polycrystalline bulk modulus B and shear modulus G are obtained using the Voigt-Reuss-Hill (VRH) approximation. Once bulk modulus B and shear modulus G are obtained, the Young's modulus E and Poisson's ratio σ can be computed 13 . The calculated values of the B, G, E and σ of NbRuB are also listed in Table 3. The B is a measure of the resistance against volume change imposed by the applied pressure, while the G denotes the resistance against the reversible deformations upon shear stress. The calculated B and G of NbRuB is 293 GPa and 156 GPa, respectively, larger than that of Nb 2 RuB 2 (B = 272 GPa, G = 146 GPa) 2 . Moreover, the shear modulus G of NbRuB is below 200 GPa, indicating the relative low resistance to shape change at a constant volume. The high bulk and shear modulus of NbRuB maybe be derived from the strong covalent bonding. According to the Pugh criterion 14 , the NbRuB is ductile material because its B/G value is higher than 1.75. Indeed ductile materials are generally metallic even though some metals can be brittle. In a word, the large elastic modulus and low Poisson's ratio show that the NbRuB would be potential hard materials.
The Vickers hardness (H v ), the intrinsic resistance to deformation when a force is applied, is another interesting property of materials. The Vickers hardness H v of complex crystals should be calculated by a geometric average of all bonds 15 : where H v u is the hardness of binary compound composed by u-type bond, P u is the Mulliken overlap population of the u-type bond, v b u is the volume of a bond of type u, d u is the bond length of type u, and N b v is the bond number of type vper unit volume. The calculated volume, bond parameter, and hardness of NbRuB crystals are presented in Table 4. It is found that the Vickers hardness value of polycrystalline NbRuB is 15.80 GPa, which is slight bigger than that of RuB 1.1 (14.5 GPa). The higher level of hardness of the NbRuB is attributed to its faint metallicity, wherein the PDOS of NbRuB at the Fermi level is much smaller than that of RuB 1.1 . As well known, the hardness is related to the elastic and plastic properties of the material. The Vickers hardness of NbRuB is also can be estimated by the Chen's empirical model. The calculated Vickers hardness of NbRuB from Chen's empirical model is listed in Table 4. It is found that the Vickers hardness is 15.29 GPa, which is close to 15.80 GPa based on expression (3)(4)(5) indicate that NbRuB is hard materials (H v > 10 GPa).
Elastic anisotropy. Elastic anisotropy is an important physical property of materials and plays a vital role in technological and industrial applications. To better describe the features of elastic anisotropy, the C ij (GPa)     are the directional cosines of angles with the three principal directions. For a perfectly isotropic material, the 3D curved surface exhibits a spherical shape, while the deviation degree from the spherical shape indicates the anisotropic character of the crystal. The obtained 3D curved surface for NbRuB is shown in Fig. 3 (a). It demonstrates that the NbRuB exhibit a strong anisotropic character in Young's modulus. The projections on the xy, xz and yz planes show more details regarding the anisotropic properties of the Young's modulus. The 2D projections of Young's modulus in those planes are shown in Fig. 3 (b). From the Fig. 3 (b), we can see that the E min = 188.8 GPa (188.8 and 319.0 GPa) and E max = 466.6 GPa (495.8 and 392.8 GPa), the ratio E max /E min = 2.47 (2.63 and 1.23) in the xy, yz and xz planes, respectively, indicating the Young's modulus of the xz plane has a weak anisotropic character compared to the other planes.
To further investigate the anisotropy, we calculated the universal anisotropic index A U 18 . For an isotropic material, A U is equal to zero, the deviation of A U from zero indicates the presence of elastic anisotropy. The calculated A U is collected in Table 3. It is noted that the NbRuB is significantly anisotropic.   Thermodynamic properties. The Debye temperature is the temperature of the highest normal mode of vibration in a crystal, and it provides insight into the thermodynamics of the material. However, the Debye temperature (θ D ) is not a strictly determined parameter, various estimates may be obtained through well established empirical or semi-empirical formula. In this work, we calculated the Debye temperature through elastic constants base on the Anderson's equation 19 . The calculated values of the Debye temperature are presented in Table 3. The Debye temperature θ D is determined to be 587.6K, which is larger than that (θ D = 468 K) 7 obtained from fitting the capacity to the Debye model in the low-temperature limit. These differences are not unexpected since the values of the Debye temperatures obtained from different definitions/experiments are expected to be different although close. From the value of DOS at E f obtained by theory calculate and experiment measure, we have estimated the value of electron-phonon coupling 20 . The superconducting transition temperature T c has been estimated using the McMillan formula 21 is the Coulomb pseudopotential, which characterizes the strength of the electron-electron Coulomb repulsion 22 . The calculated values of λ, µ ⁎ and T c are given in Table 3 along with the experimental data. In our work, the calculated µ ⁎ = 0.16, λ = 0.584, the estimated T c = 4.5 K, which is larger than the experimental data T c = 3.1 K. It should be considered that the phonon calculation was not performed with a shifted set of bands, and given that the effect of that shift was to decrease N(E f ). As seen from Fig. 2, the contribution of the 4d-state is larger than those of the 5s and 4p states. The larger contribution of 4d state electrons clearly shows the possibility of superconductivity in NbRuB.

Conclusions
In summary, the electronic structure, chemical bonding, mechanical and thermodynamic properties of NbRuB have been studied via first-principles. The optimized lattices parameters are slightly underestimate the experimental data. The DOS reveals that NbRuB exhibit metallic behavior, and the valence bands stems from 4d states of Ru and Nb, the conduction bands originate mostly from Nb-4d and B-2p states. The Mulliken charge population shows this compound has mixed covalent-ionic property. The elastic properties of this material are analyzed, and the results show that NbRuB is mechanically stable and an elastic anisotropy. The calculated B/G ratio and Vickers hardness shows that NbRuB is ductile and hard material. Moreover, the values of the superconducting temperature T c of NbRuB is predicted to be 4.5 K, which is larger than the experimental value T c = 3.1 K. The superconductivity of NbRuB may be related to its electronic properties and the geometry structure.