DENSITY FUNCTIONAL PREDICTION OF THE STRUCTURAL, ELASTIC, ELECTRONIC, AND THERMODYNAMIC PROPERTIES OF THE CUBIC AND HEXAGONAL (c, h)-Fe2Hf‚

Using density functional theory (DFT), the structural, elastic, electronic, and thermodynamic properties of Fe2Hf in the cubic and hexagonal solid phases with Fd-3m and P63/mmc are reported with generalized gradient approximations (GGA). To achieve energy convergence, we report the k-point mesh density and plane-wave energy cutoffs. The calculated equilibrium parameters are in good agreement with the available theoretical data. A complete elastic tensor and crystal anisotropies of the ultraincompressible Fe2Hf are determined in the wide pressure range. Finally, by using the quasi-harmonic Debye Model, the isothermal and adiabatic bulk modulus and heat capacity of Fe2Hf are also successfully obtained in the present work. By the elastic stability criteria, it is predicted that Fd-3m and P63/mmc structures of Fe2Hf are stable in the pressure range studied, respectively.


Introduction
Up to now, Fe2Hf structures are not still synthesized in crystalline form. We have proposed two structures of cubic and hexagonal type structures. Elastic constants under pressure play an important role in determining the strength and hardness of materials. They are very important to determine the response of the crystal to external forces, as characterized by the bulk and shear modulus. The elastic properties of two cubic and hexagonal structures are also explored. However, to our knowledge, no works have reported on the elastic, structural, and thermodynamic properties of hexagonal structures of Fe2Hf under pressures. Koki Ikeda et al. [1] studied the crystal structures and magnetic properties of the stoichiometric Laves phase in the iron hafnium system by X-ray powder techniques and magnetization measurements. They concluded that the hexagonal MgZn2-type Fe2Hf compound occurs above 1673 K, and the cubic MgCu2-type exists below 1273 K.

Computational details
CASTEP was the first-principles plane-wave pseudo-potential method based on the density functional theory, the crystal wave functions launched by Plane Wavelet Group using periodic boundary conditions, in which the exchange-correlation potential adopted GGA. Without taking into account the spin polarized, we must first investigate and examine the convergence of calculated total energies with respect to the plane wave cut-off Ecut in the cubic and hexagonal solid phases of the two molecular systems to be 380eV (385eV) for c-Fe2Hf (h-Fe2Hf) Fig.1 (a) and (b) respectively. The number of points can be arbitrarily increased to increase the precision of calculations but this increases the computational cost.

Structural properties
In this work, we focus on structural, elastic, electronic, and thermodynamic properties of Fe2Hf with cubic (hexagonal) structures under pressures up to 10GPa (15GPa) by first-principles calculations. The elastic properties of two cubic and hexagonal structures of Fe2H funder pressure are investigated for the first time, from which the mechanical stability is also determined. In order to further investigate Fe2Hf the thermodynamic properties, such as the heat capacity, thermal expansion, Debye temperature, and so on, are obtained by Debye model. However, it is difficult to study experimentally because of the extremely small sizes and time scales at which the functional properties appear. Theoretical modeling, on the other hand, offers a way to overcome these difficulties through "virtual experiments" that can enable us to quickly, efficiently, and affordably explore phase space. Density functional theory (DFT) is one of the most successful quantum mechanical theory based tools to characterize properties of materials, and it is widely used in quantum computation in condensed matter physics [6][7][8]. DFT is a general-purpose computational method and can be applied to most systems. Although density functional theory is more accurate and exact in theory, its implementation requires several approximations such as the choice of basis-set, exchange-correlation [9][10][11] functional, mesh-size for Brillouin zone [12][13][14] for planewave basis.
The plane wave cut-off is the parameter that controls such truncation, so the more include plane waves; the better the wave function is modeled. The k-point mesh controls the BZ integration, can play a huge role in the quality of the results, and heavily depends on the number of these points on the mesh-grid, especially for metals. The plane wave basis set in the DFT wave function is expanded in terms of a: due to its periodicity and ease of use. Each function ui.k(r) is expressed as a Fourier series whose basis states are plane waves whose wave vector is a reciprocal lattice vector G (which are defined by e iG•R = 1).
The DFT wave function is expanded in terms of a plane wave basis set: where, G is the reciprocal wave vector, k is k-point vector, CG are the expansion coefficients. There is an infinite number of allowed G, but the coefficient CG becomes smaller and smaller as G 2 becomes larger and larger. The cut-off energy Ecut is defined [15] as: Any standard DFT code such as CASTEP uses the total energy convergence rather than energy per atom during an SCF procedure. Fig.1; represents total energy as a function of the number of k-points. The convergence has been achieved in the k-points sampling of 1x1x1 for total energy convergence tolerance (1.0 10 -6 eV/atom).
The rest of this paper is structured as follows. The computational method is described in section 2. The results and some discussions of structural, elastic, and thermodynamic properties of Fe2Hf under pressure are presented and compared with available experimental and theoretical data in section 3. Conclusions derived from our calculations are drawn in section 4. The large and small balls represent Fe and Hf atoms, respectively, as shown in Fig. 2. Secondly, we examine the convergence of the calculated number of points of our computed compounds that are 6x6x6 of c-Fe2Hf and h-Fe2Hf, Fig. 3. In Fig. 4, we show the plot of energy versus volume of Fe2Hf in cubic (a) and hexagonal (b) structures. The density functional calculations on the structural, elastic, and thermodynamic properties of cubic and hexagonal structures of Fe2Hf were performed using the CASTEP code [16,17]. Both lattices were optimized to get the equilibrium structure for cubic and hexagonal structures of Fe2Hf. The exchangecorrelation function was treated by both the generalized gradient approximation (GGA-PBE) [18]. Also, the pseudo potentials constructed using the ab initio ultra-soft scheme to describe the valence electron interaction with the atomic core, in which the Fe (3d 6 4s 2 ) and Hf (4f 14 5d 2 6s 2 ) orbital are treated as valence electrons. The Brillouin zone sampling was carried out using the 6x6x6for cubic and hexagonal structures of Fe2Hf set of Monkhorst-Pack mesh [13]. The structures were optimized using the Broyden-x Fletcher-Goldfarb-Shenno (BFGS) minimization technique in order to achieve the most stable structure in their local area. Self-consistent convergence condition setting: the total energy was less than 0.2 eV/atom, the force in each atom was less than 0.05eV/Å, the offset tolerance was less than 0.0002 A, and the Stress bias was less than 0.1GPa. For comparison and as shown in Table 1, it should be noted that the calculated total energies of cubic and hexagonal (c-h)-Fe2Hf structures are with a negative value corresponding to an exothermic reaction. The total energies show that the hexagonal structures are more stable than cubic ones.

Elastic constants
By means of a Taylor series expansion of the total energy, E (V, δ), the elastic constants are defined for the system compared to a small deformation δ of the lattice unit cell volume V. The energy of system constraint is expressed as follows [19]: Where E(V0, 0) is the energy of the unstrained system with equilibrium volume V0, τi is an element in the stress tensor, and ξi is a factor to take care of the Voigt index. Between 21 and 2 independent elastic constants, corresponds respectively to asymmetric material and isotropic material, the number of independent elastic constants is 3 for cubic and 5 for hexagonal crystals. These three independent elastic constants are usually referred to as C11, C44, and C12 for cubic and C11, C33, C44, C12, and C13 for hexagonal crystals. A theoretical treatment of the elasticity of hexagonal systems is thus considerably more involved than for cubic structure, which has three independent elastic constants. The elastic constants of solids provide a link between mechanical and dynamical behaviours of crystals and give important information concerning the nature of forces operating in solids. In particular, they provide information on the stability and stiffness of materials. It is well known that first-order and second-order derivatives of the potential give forces and elastic constants. Therefore, it is an important issue to check the accuracy of the calculations for forces and elastic constants. Let us recall here that the pressure effect upon elastic constants is essential, at least for understanding interatomic interactions, mechanical stability, and phase transition mechanism. The corresponding bulk moduli are determined as a function of pressure up to 50 GPa for cubic and hexagonal Fe2Hf structures. Let us notice that all elastic constants, as well as both bulk moduli, linearly increase when pressure is enhanced Fig. 5. The bulk modulus was obtained from a calculation of the three cubic elastic constants 11 , 12 and 44 using the expression: The calculated bulk value with the second method is agreeable with the bulk modulus, which is deducted from the first method. For Fe2Hfin, a cubic crystal, the generalized elastic stability criteria in terms of elastic constants [20]: For Fe2Hf in a hexagonal crystal, let us recall that the generalized elastic stability criteria [21] are: 11 > 0, 33 > 0, 44 > 0, 66 > 0 , 11 − 12 > 0, 11 + 33 + 12 > 0, ( 11 + 12 ) 33 − 2 13 2 > 0 8 x The fact that the elastic constants of (Fe2Hf) in the cubic [hexagonal] crystal do not obey all the above criteria indicates that it is an elastically unstable [stable] structure. For Fe2Hfwith hexagonal structure, the shear anisotropy factor A(Cij), defined as A=4C44/(C11+C33-2C13) = 0.88 for the {100} shear planes between the < 011> and< 010> directions, [22] as well as the ratio between linear compressibility coefficients for hexagonal crystals, i.e. kc/ka= (C11+C12-2C13)/(C33-C13) = 1.22.
For hexagonal structures, bulk modulus B and shear modulus G are obtained as follows: We can conclude that the bulk modulus is 219 GPa for cubic Fe2Hf and 222 GPa for hexagonal one. All values are exceptionally high, exceeding or matching other hard materials, including boron carbide (B4C, 200 GPa), silicon carbide (SiC, 248 GPa), sapphire (Al2O3, 252 GPa), and cubic boron nitride (c-BN, 367 GPa) [24]. Pugh [25] proposed the B/G ratio to represent a measure of a "machine able behavior". A high B/G (2.44 for hexagonal Fe2Hf type structure) value is then associated with ductility and a low value with brittleness. The critical value which separates ductile and brittle behaviors is at about 1.75. For example, diamond has a B/G of 0.80 [26], while aluminum, cobalt, rhodium, and iridium present B/G ratios of 2.74, 2.43, 1.77, and 1.74, respectively [25]. The B/G calculated ratios for the hexagonal structure ofFe2Hfcompound, we obtain is of 2.44. The fact of Fe2Hf compounds is ductile compound. Additionally, the mechanical stability of the hexagonal structure at 0 GPa can be predicted from the elastic constants data (C11 -C12> 0). To be complete, the elastic constants of pure Fe2Hf in cubic and hexagonal structures are listed in Table 1. The Debye temperature may be estimated from the average sound velocity Vm [26].
where h is Planck's constants, k is Boltzman's constant, NA is Avogadro's number, n is the number of atoms per formula unit, M is the molecular mass per formula unit, ρ is the density, and Vm is obtained from [27].
where Vs and Vl are the shear and longitudinal sound velocities, respectively. The arithmetic average of the Voigt and the Reuss bounds is called the Voigt-Reuss-Hill (VRH) average and is commonly used to estimate elastic moduli of polycrystals. The VRH averages for shear modulus (G) and bulk modulus (B) are; The polycrystalline moduli are the arithmetic mean values of the Voigt and Reuss moduli [28]: Therefore, the probable values of the average shear and longitudinal sound velocities can be calculated from Navier's equation [29]: The longitudinal, transverse and average sound velocities and Debye temperature of cubic and hexagonal of Fe2Hfin cubic and hexagonal structures have been calculated and listed in the Table 1.
At zero pressure and zero temperature, we obtain, as shown in Table 1, we have calculated the sound velocities and Debye temperature for the Fe2Hf compounds from our elastic constants. Our calculated sound velocities and Debye temperature are comparable to the experimental values.

Electronic structure
In this part, the electronic properties of Fe2Hf in a cubic and hexagonal structure are discussed. Fig. 6 shows the total density of states (DOS) for our compound in cubic and hexagonal structures at equilibrium lattice constants. We show here only the vicinity of the Fermi energy level. Fe2Hf in cubic [hexagonal] structure have similar DOS profiles in the whole energy region except for some differences. On the other hand, near the Fermi level, the DOS mainly originates from the M-d bands M (Fe2Hf) in cubic and hexagonal structure, this suggests that our compounds are all conductive, and the d bands of the transition metal play the dominant role in electrical transport, this phenomenon is very pronounced especially in the hexagonal structure. The DOS at the Fermi level n(EF) is calculated to be 10.69 states/eV unit cell for cubic Fe2Hf and 20.11 eV for hexagonal of Fe2Hf in cubic and hexagonal structure indicating the metallic material. Generally speaking, the smaller n(EF) is, the more stable the compound is. So in agreement with the results obtained from the total energy minimum.

Thermodynamic properties
A study of the effects of chemistry and crystal structure must be performed for motors with operating temperatures in the region of 2000 °C that will require materials that can withstand. Intermetallic compounds with high melting temperatures are candidates for these applications. Thermodynamics is one of the great theories on which the current understanding of the material is based; thermodynamics is mainly based on temperature and entropy, which is the degree of disorganization of the material. Physical properties under pressures and temperatures have important meanings to accelerate the understanding and synthesis of (c, h)-Fe2Hf structures. The investigation of the thermal capacity of crystals is an interesting subject in solid state physics because it enters many applications and provides essential information on its vibratory properties. According to the standard theory of elastic continuum, two limiting cases are correctly predicted. At sufficiently low temperatures, the thermal capacity CV is proportional to T 3 . At high temperatures, CV tends to the limit Petit and Dulong. Applying the quasi-harmonic Debye model to the (c, h)-Fe2Hf structures, we calculated the thermal capacity of the , network and the Debye temperature at different temperatures. Now we investigate the dependences of bulk modulus B on temperature T and pressure P. B is plotted in Fig. 7.   Fig. 9. From the quasi-harmonic Debye model, we obtained the Debye temperature = 415.5 K and 458 K at P=0 GPa and T=0 K for respectively c-Fe2Hf and h-Fe2Hf structures. The heat capacity CP and CV, represents the heat absorbed by the crystal at constant pressure or constant volume necessary to raise the temperature of one mole of a pure substance by one-degree K generated by this transformation. The heat capacity of a crystal is given by a relation deduced from the vibratory motions of the crystal lattice it is also mandatory for many applications. For solids and liquids, the variation of the PV product  8-10. From this figure, one can see the sharp increase of CP and CV in the temperature range from 0up to ~500 K, and at high temperature, the CP and CV tend to a constant value (300 J·mol -1 K -1 ) [150 J·mol -1 K -1 ] for (c-h)-Fe2Hf the so-called Dulong-Petit limit of 3nkB value [30]. We can conclude that the volume heat capacity in 300°K is in the range of 120-150 J·mol -1 K -1 for c-Fe2Hf and 250-300 J·mol -1 K -1 for h-Fe2Hf. All values in this range are exceptionally high, exceeding or matching other hard materials, including silicon oxide (SiO2, 71.5 J·mol -1 K -1 ), sapphire (Al2O3, 120.5 J·mol -1 K -1 ), and (Cr2O3, 125.1 J·mol -1 K -1 ) and approaching that of diamond (B0 = 442 GPa).   Fig. 12, the entropy function of the temperature at different pressures and of the pressure at different temperatures of the two types of c-Fe2Hf and h-Fe2Hf structures of the compound. We show that the entropy increase parabolically with increasing temperature. We notice that these curves have the same shape except the existence of a slight offset at high temperatures especially for the case of the hexagonal type (upper panel). In the lower panel we show the variation of the entropy function S, as a function of the pressures and at different temperatures for the two types of c-Fe2Hf and h-Fe2Hf structures, we notice that these curves are similar and that the entropy function S is significant in the case of the hexagonal type.

Conclusion
In summary, in this work, we theoretically obtain results for structural, elastic, mechanical, electronic, and thermodynamic properties of Fe2Hf in the cubic and hexagonal solid phases, based on the ab initio total energy calculations. The estimated lattice constants are in excellent agreement with the available experimental values. The total energies show that the hexagonal structures are more stable than cubic ones. The DOS at the Fermi level n(EF) is calculated to be 10.69 states/eV unit cell for cubic and 20.11 eV for hexagonal, indicating the metallic material. Generally speaking, the smaller n(EF) is, the more stable the compound is. So in agreement with the results obtained from the total energy minimum. The analyses of bulk modulus indicate that all values calculated are exceptionally high, exceeding or matching other hard materials, including boron carbide (B4C, 200 GPa), silicon carbide (SiC, 248 GPa), sapphire (Al2O3, 252 GPa), and cubic boron nitride (c-BN, 367 GPa). Unfortunately, for the other computed properties in this work, there are no previous calculations or experimental values to compare with.