Nonlinear Elasticity of Borocarbide Superconductor YNi2B2C: A First-Principles Study

First-principles calculations combined with homogeneous deformation methods are used to investigate the secondand thirdorder elastic constants of YNi2B2C with tetragonal structure. The predicted lattice constants and second-order elastic constants of YNi2B2C agree well with the available data. The effective second-order elastic constants are obtained from the secondand thirdorder elastic constants for YNi2B2C. Based on the effective second-order elastic constants, Pugh’s modulus ratio, Poisson’s ratio, and Vickers hardness of YNi2B2C under high pressure are further investigated. It is shown that the ductility of YNi2B2C increases with increasing pressure.


Introduction
Recently, a considerable number of superconducting binary, ternary, and quaternary Ni-based systems are reported [1] and Ni element is ferromagnetic; the identification of superconductivity in Ni-based compounds is of high current interest.Cava et al. [2] have synthesized the quaternary nickel borocarbides RNi 2 B 2 C (R=Y and rare earths) and found that Lu and Y compounds exhibit the highest superconducting transition temperatures (  = 16.6 and 15.6 K, resp.), while compounds with magnetic rare earths exhibit lower   :   (  = 11 K),   (  = 10.5 K), and   (  = 8 K).Because RNi 2 B 2 C [3] has the elevated superconducting transition temperature for some of the members of the family (R = Lu, Y) or the interesting interaction between superconductivity and magnetic ordering phenomena for the compounds with R = magnetic rare earth, considerable theoretical as well as experimental attention has been focused on these quaternary Ni-based compounds.Early work reported that single crystal YNi 2 B 2 C, one of the quaternary intermetallic compounds, is a traditional and isotropic superconductor.Therefore, a lot of work including theory and experiment has been done to examine various properties of YNi 2 B 2 C [4][5][6][7][8][9][10].For example, Godart el al. [11] systematically investigated the structural, superconducting, and magnetic properties of YNi 2 B 2 C by X-ray diffraction measurements.Lee et al. [12] systematically researched the electronic structures of Nibased superconducting quaternary compounds YNi 2 B 2 X (X = B, C, N, and O) by employing the linearized muffin-tin orbital band method.Meenakshi et al. [13] investigated the high pressure behavior of YNi 2 B 2 C at room temperature by electrical resistivity, thermopower, and X-ray diffraction incorporating imaging plate.More recently, Wang et al. [3] calculated the elastic and electronic structure properties of YNi 2 B 2 C under pressure by performing the generalized gradient approximation (GGA) and local density approximation (LDA) correction scheme in the frame of density functional theory (DFT).All these researches are very important to the further scientific and technical investigations.
At it is well known, elastic constants of a solid are important since some physical properties such as the bulk modulus, shear modulus, Young's modulus, and Poisson's ratio can be derived from the elastic constants and the strength of materials and the velocity of sound of longitudinal wave and shear wave can be also determined from its elastic constants [14].In the finite-strain theory of elastic deformation, the secondorder elastic constants (SOECs) are sufficient to describe the linear elastic stress-strain response [15].In nonlinear elastic theory, high-order elastic constants, such as third-order elastic constants (TOECs), play an important role as well as SOECs [16].TOECs not only are useful in describing the mechanical response of crystals under high stress and strain, but also serve as a basis for describing anharmonic properties such as thermal expansion, phonon-phonon interaction, and Grüneisen parameter [17,18].Though many experiments have been performed to determine SOECs and high-order elastic constants [19], to obtain a complete set of TOECs is still important.However, the TOECs are very difficult to measure experimentally.Recently, a simple method using first-principles calculations has been employed to determine TOECs [20][21][22], and their results show good agreement with experiments.So far, as we know, the nonlinear elasticity as well as TOECs in YNi 2 B 2 C has not been reported in the literature.To understand the physical properties of YNi 2 B 2 C and provide significant information with respect to applications of YNi 2 B 2 C, it is very necessary to study the nonlinear elastic properties.In this paper, the complete set of TOECs for YNi 2 B 2 C are presented from first-principles calculations combined with the method of homogeneous deformation, and the effective SOECs are also obtained.

Theory and Computational Details
Our method for calculating elastic constants mentions homogeneous continuum elasticity theory [23][24][25].After applying a finite homogeneous deformation to a material, the initial configuration   at the equilibrium will move to the final configuration    .The deformation gradient is described as follows: From the deformation gradients, we may define the Lagrangian strain The elastic energy (Δ) can be expanded in a Taylor series in terms of the strain tensor: where  is the volume of the unstrained lattice.When the elastic energy Δ is the internal energy  and the Helmholtz free energy , the elastic constants represent the isentropic and isothermal, respectively.Since our first-principles calculations are performed at 0 K,  =  −  = , the isentropic elastic constants are equal to the isothermal elastic constants.After using the Voigt notation for the strain tensors  11 → For YNi 2 B 2 C with tetragonal structure, there are six independent SOECs and twelve independent TOECs.The number of applied strain tensors must be as large as the number of independent TOECs for solving the TOECs.To obtain all the SOECs and TOECs of YNi 2 B 2 C, we select twelve simple deformation modes such that the strain tensor only has one or a few components.Any nonzero component of each strain tensor is denoted by a single scalar parameter .After inserting various strain tensors into the above strain energy, we can find that the strain energy can be expressed as a polynomial function of the strain tensor parameter ; namely, where the coefficients Λ 2 and Λ 3 are of SOECs and TOECs of the crystals, respectively.The selected twelve strain tensors   ( = 1, 2, . . ., 12) and the corresponding coefficients Λ 2 and Λ 3 are presented in Table 1.For each strain tensor   , the strain parameter  varies from −0.05 to 0.05 with a finite step size 0.005.We have performed first-principles calculations based on the DFT level, using the VASP code developed at the Institut für Materialphysik of Universität Wien [26][27][28].The ultrasoft pseudopotentials are employed to describe the electron-ion interactions.To compare the performance of different approximations of exchange-correlation interaction, in the structure calculations of YNi 2 B 2 C, both the GGA proposed by Perdew and Wang [29] and the LDA proposed by Vosko et al. [30] are considered as the exchange-correlation potential.For the elastic constants calculation, the effects of exchangecorrelation interaction are treated with the GGA.For the Brillouin zone (BZ) integrals, reciprocal space is represent by the Monkhorst-Pack special -point scheme [31].Since high accuracy is needed to calculate the TOECs, the -point mesh size with 15 × 15 × 5 and cutoff energy with 550 eV are used to calculate the lattice and elastic constants.The totalenergy of electronic self-consistency is converged to 10 −6 eV.All the calculations avoid wrap-around errors and use an augmentation grid that is exactly twice as large as the coarse grid for the representation of the pseudo wave functions.

Results and Discussion
3.1.Structure Parameters.For the tetragonal structure YNi 2 B 2 C, the initial structural model is built according to previous available lattice parameters  and .The tetragonal YNi 2 B 2 C has 4/ space group and its crystal structure is shown in Figure 1.In a unit cell, the atom of Y is set at position (0, 0, 0), the two Ni atoms are set at positions (0, 0.5, 0.25) and (0, 0.5, 0.75), the two B atoms are set at positions (0, 0, ) and (1, 1, 1 − ), and the C atom is set at position (0, 0, 0.5).We optimize the structural parameters; full relaxations are carried out with respect to the volume, shape, and all internal atomic positions for the unit cell.In Table 2, we list our results together with the available experimental data [10,11,32,33] and other theoretical results [1,3].The optimized lattice parameters are  = 3.543 Å,  = 10.428Å from the GGA method and  = 3.478 Å,  = 10.188Å from Table 1: Selected strain tensors and the coefficients Λ 2 and Λ 3 in equation ( 5) as the linear combinations of the second-and third-order elastic constants for tetragonal crystal.

Strain
Λ 2 Λ 3  1 = (, 0, 0, 0, 0, 0)  11  111  2 = (, , 0, 0, 0, 0) 2  the LDA method.The results with GGA are slightly larger than those with LDA.Comparing with the results with LDA, the results with GGA are closer to the previous results.We also obtained the crystal coordinate  = 0.358 from the GGA method, which agrees well with the results by Wang et al.
[3], Ravindran et al. [1], and Siegrist et al. [32] but is far from that by Belger et al. [33].The crystal coordinate obtained from the LDA method is 0.355; the difference between GGA and LDA is small; however, the result from GGA is closer to the experimental results.In a word, from the structural parameters, it is found that the results from GGA seem to be the best.Therefore, in the following, the calculations with GGA are used to obtain the SOECs and TOECs of YNi 2 B 2 C.

The SOECs and TOECs.
The strain-energies for YNi 2 B 2 C, including the results of the first-principles calculations and the fitted polynomials, are shown in Figure 2. The discrete points denote the values from the first-principles calculations and the solid lines represent the results determined from the third-order polynomial fitting.It is worth noting that for YNi 2 B 2 C with Lagrangian strains up to 5.0%, including the terms up to third order in energy expansion sufficed to obtain good agreement with our ab initio results.In other words, the terms up to the third order in energy expansion produce the reliable results.Furthermore, we focus on examining for which range of strains the third-order effects dominate the properties of YNi 2 B 2 C. Taking  6 as an example, Figure 3 shows the curves of the linear elasticity in comparison with nonlinear elasticity as well as the first-principles calculations.It is easily found that the linear elasticity is not sufficient and the relative error between the linear elasticity and the nonlinear elasticity is larger than 11.3% in strain energy density when the applied strain is larger than about 3.0%, so the nonlinear elasticity needs to be considered.Knowledge of the values of elastic constants is crucial for understanding the structural stability, the bonding characteristic between the adjacent atomic planes, and the anisotropic character of the bonding.Table 3 gives the present SOECs as well as the experimental results [7,8] and the other theoretical data [3].For SOECs values, sometimes we possibly obtain slightly different results from different fitted curves.For example,  13 calculated from coefficients in  4 (),  5 (), and  6 () are 136.74GPa, 132.84 GPa, and 137.47 GPa, respectively, and  44 calculated from coefficients in  7 (),  8 (), and  10 () are 62.60 GPa, 65.07 GPa, and 59.32 GPa, respectively.In these cases, we give the average of all obtained values in Table 3.It is found that our values of SOECs provide good agreement with the results obtained from the previous calculations and experiments, which provides guarantee for calculating the TOECs accurately.Obviously, the calculated elastic constants of YNi 2 B 2 C obey  the following well-known mechanical stability criteria [34] for the tetragonal structure:  Thus, YNi 2 B 2 C with tetragonal structure is mechanically stable at the ground state.The elastic constants  11 and  33 are directly related to sound propagation along the crystallographic direction, while other shear elastic constants ( 12 ,  13 ,  44 , and  66 ) are dominated along the nonaxial sound propagation.The results show that it has larger   along the axial directions rather than the nonaxial directions indicating that it is more uncompressible along axial directions.We care, moreover, that  11 >  33 , which indicates that the atomic bonding strength along the [100] direction between the nearest neighbors is stronger than that along the [001] direction.
The calculated TOEC values of tetragonal YNi 2 B 2 C single crystal are summarized in Table 4. Unfortunately, there are currently no experimental and other theoretical values for comparison.It is known that the TOECs provide a comprehensive knowledge concerning the vibrational anharmonicity of the acoustic modes in long-wavelength limit.From Table 4,

DFT results
The third-order polynomial fitting The second-order polynomial fitting it is observed that the TOECs values of tetragonal YNi 2 B 2 C are calculated to be negative except for two positive values of  123 (=195.47GPa) and  366 (=111.09GPa).Furthermore, the absolute value of  111 is the greatest among the TOECs indicating the anisotropy along the basal plane to be more pronounced than along the unique axis.

Elastic Properties under
The variation of the effective SOECs    with pressure in the range of 0-5 GPa for YNi 2 B 2 C on the basis of the calculated SOECs and TOECs is shown in Figure 4.It is observed that the longitudinal mode  11 as well as the shear modes  12 ,  13 ,  44 , and  66 of YNi 2 B 2 C can increase linearly; in contrast the longitudinal mode  33 decreases linearly with increasing the pressure. 11 and  13 move up very sharply, and  12 ,  44 , and  66 have the same growth rate.For a tetragonal crystal, the mechanical stability leads to restrictions on the elastic constants under pressure as follows [36] 5(a) shows the pressure dependence of  and  for YNi 2 B 2 C. It can be seen from Figure 5(a) that bulk modulus increases but the shear modulus reduces with increasing pressure, which indicates that the ductility of YNi 2 B 2 C increases with increasing pressure.Based on the bulk modulus and the shear modulus, Pugh [38] has proposed that the quotient of / can predict the brittle and ductile behavior of materials, and thus a low value of / is associated with ductility while a high value is associated with brittleness.All the values of Pugh ratio / in the pressure range of 0-5 GPa are showed in Figure 5(b).Obviously, all the values of / decrease with pressure which means that pressure can improve ductility.In addition, Poisson's ratio with ] = (3−2)/2(3+) can also reflect the ductile properties.Poisson's ratio is reversely proportional to / [39]; that is to say, the larger the Poisson ratio, the better the ductility.Obviously, the hardness of YNi 2 B 2 C decreases with increasing pressure.Besides, the low / ratio corresponds to poor hardness.Therefore, the ductility is enhanced with the decreasing of hardness.

Conclusions
In the present work, the SOECs and TOECs of tetragonal YNi 2 B 2 C are systematically investigated using first-principles methods combined with finite-strain theory.In comparison with the linear elastic theory, the nonlinear elastic effects must be considered when the applied strain is larger than about 3.0%.In terms of the SOECs and TOECs, the effective SOECs of YNi 2 B 2 C are presented.By the elastic stability criteria under isotropic pressure, it is predicted that YNi 2 B 2 C with tetragonal structure is mechanically stable in the pressure range of 0-5 GPa.Based on the effective SOECs, Pugh's modulus ratio, Poisson's ratio, and Vickers hardness of YNi 2 B 2 C under high pressure from 0 to 5 GPa are further investigated.Pugh's modulus ratio and the Vickers hardness decrease and Poisson's ratio increases with increasing pressure, resulting in a decrease of the hardness and an increase of the ductility of YNi 2 B 2 C.

Figure 2 :
Figure 2: The strain energy relations for tetragonal YNi 2 B 2 C. The discrete points denote the values from first-principles calculations and the solid curves represent the results obtained from the third-order polynomial fitting.

Figure 3 :
Figure 3: Energy as a function of the Lagrangian strain parameter  for the particular strain tensor  6 .The circles indicate the values of the first-principles calculations; solid and dashed curves represent the results obtained from nonlinear and linear elasticity theory, respectively.

Figure 5 (
b) also presents Poisson's ratio of YNi 2 B 2 C as a function of pressure.It is clearly shown that Poisson's ratio has the opposite change tendency with /; that is, we can obtain the same conclusion from the change of Poisson's ratio and /; namely, the ductility of YNi 2 B 2 C increases with increasing pressure.As we all known, both Pugh's modulus ratio and Poisson's ratio have close relationship with the hardness of materials.Hardness is another important parameter to describe the mechanical properties of a material.The formula of the Vickers hardness proposed by Chen et al. [40, 41] can be expressed as  V = 2( 2 ) 0.585 − 3, where  = /.The change in Vickers hardness versus pressure for YNi 2 B 2 C is shown in Figure 5(b).

Table 2 :
Calculated lattice parameters at  = 0 K and  = 0 GPa compared with the experimental data and other theoretical results.

Table 3 :
[8]ond-order elastic constants   (GPa) of tetragonal YNi 2 B 2 C at 0 K and 0 GPa, in comparison with the previous calculations and available experimental measurements.Calculated from the quoted values of  11 and ( 11 −  12 )/2[8]. a
Pressure.It is helpful to describe the effective SOECs under different pressure using the SOECs and TOECs.The effective SOECs    of a crystal can be obtained when subjected to pressure .For YNi 2 B 2 C with tetragonal structure, the expressions for the six effective SOECs are as follows [35]:   11 =  11 +  (4 11 + 2 12 +  111 +  112 ) +  (− 11 + 2 13 +  113 ) ,   12 =  12 +  (2 12 + 2 112 ) +  (− 12 + 2 123 ) ,   13 =  13 +  ( 113 +  123 ) +  ( 13 +  133 ) , It is obvious that the elastic constants of YNi 2 B 2 C satisfy all of these conditions in the pressure range of 0-5 GPa.The bulk modulus  represents the resistance of a material to volume (bond-length) change under hydrostatic pressure, and the shear modulus  describes the resistance to shape Substituting the values of the effective SOECs    into the bulk modulus  and shear modulus  expressions, we can obtain the values of  and  under different pressure.Figure