3.1 Structural properties and formation energy

The Laves phase LaCo₂ compound investigated in the present work crystallizes in the face centered cubic MgCu₂ type structure with space group Fd-3m (No. 227). The La atoms occupy the tetrahedral 8a Wyckoff site(1/8,1/8,1/8) and the Co atoms occupy the octahedral 16d site(1/2,1/2,1/2) [¹⁸], as shown in Fig. 1. We used the experimental lattice constant to start the calculations. The optimized lattice parameters a, cell volume V , bulk modulus B, and the first pressure derivative of the bulk modulus B’ recorded in Table I, are determined by fitting the total energy per cell with the Birch-Murnaghan equation of state [¹⁹,²⁰]. The Energyoptimization curve of LaCo₂ using GGA approximation is also shown in Fig. 2. The calculated lattice constant of LaCo₂ intermetallic is 7.274 Å so we can deduce that the experimental and theoretical values are in good agreement.

FIGURE 1 Crystal structure of Laves Structure (C15) LaCo₂. 

FIGURE 2 Total energy computed against volume for LaCo₂ in the C15 structure. 

We therefore measured the formation energy for LaCo₂ intermetallic compound in order to study the relative phase stability. The formation energy (∆E) can be calculated by the following expression [²¹]:

where ELaDHCP and ECoHCP are the total energies of the elements La and Co atom in their stable bulk structures, and E _LaCo2 is the equilibrium total energy of the LaCo₂ compound. The estimated formation energy of LaCo₂ is included in Table I, it gives the value of −0.6211 (meV) which is negative indicating that this phase is stable from an energy point of view.

3.2 Mechanical stability and elastic properties

The mechanical stability of materials under the effect of hydrostatic pressure has a great influence on their performance in industrial and technological applications. Furthermore elastic stiffness constants of solids are very important material parameters because they provide a connection between mechanical properties and dynamic information on the nature of the forces operating in solids, in particular for the stability, stiffness, brittleness, ductility, material anisotropy and the propagation of elastic waves in the normal material mode [²²].

It is also essential to perform the calculations of the pressure dependence of the elastic constants for LaCo₂. Since the intermetallic compound corresponds to the cubic crystal, it has three independent elastic constants C₁₁, C₁₂ and C₄₄. Table II displays the determined elastic constants of LaCo₂ at pressures of up to 90 GPa. We believe that this is the first work that shows the elastic constants of LaCo₂ under pressure, and there are no comparable studies in the literature. So our study should encourage the experimental confirmation of these results in future.

From Table II, the measured values of the elastic constants under ambient conditions properly follow the cubic stability criteria [²³]: C₁₁ > 0, C₄₄ > 0, (C₁₁ - C₁₂) > 0, (C₁₁ + 2C₁₂) > 0, C₁₂ < B < C₁₁. Our LaCo₂ compound in the C15 structure is therefore mechanically stable. But these stability criteria are only valid in the case of zero pressure [²⁴]. For a cubic material under hydrostatic pressure, the generalized stability criteria are summarized as follows:

Figure 3 represents the variation of the generalized stability criteria Mi, as a function of the hydrostatic pressure in an interval of 0 to 90 GPa. According to our calculations, the Mi values obtained increase with increasing pressure, which means that the LaCo₂ compound keeps its rigidity under this pressure interval.

FIGURE 3 The generalized stability criteria M _i of the LaCo₂ compound as a function of pressure. 

An estimation of ductility and brittleness can be provided from the Cauchy pressure value (C₁₂−C₄₄), if it is negative the material is expected to be brittle and positive values as ductile. Furthermore the Cauchy pressure is used to define the angular character of atomic bonding in metals and compounds [²⁵]; a negative value means that the material is nonmetallic with directional bonding and the material is supposed to be metallic if the Cauchy pressure is positive [²⁶]. From Table III, it is evident that the values of the Cauchy pressures of LaCo₂ at ambient condition and under all the applied pressures are positive approving the ductility and the metallic behavior of this compound.

The Pugh’s ratio (B/G) [²⁷,²⁸] is an another stability criterion to determinate whether the material is ductile or brittle. If B/G > 1.75 the material is ductile or otherwise brittle. It is evident from Table III that LaCo₂ compound is ductile at ambient pressure and that the ductility decreases gradually with increasing pressure. This agrees well with the Cauchy pressure estimation.

The polycrystalline elastic moduli can be computed using the Voigt-Reuss-Hill (VRH) method [29], based on the elastic constants. The appropriate equations are given as follows for the cubic structure:

Here, B, G and E are the polycrystalline bulk modulus, shear modulus and Young’s modulus, respectively. v is the Poisson ratio.

Figure 4 shows the pressure dependence of bulk, shear, and Young’s moduli of C15-type LaCo₂. It is clearly seen that the value of B, which represents the resistance of the crystal to the volumetric change, increases with the increase of the pressure, which means that the presence of external pressure increases resistance to deformation.

FIGURE 4 The pressure dependence of bulk, shear, and Young’s moduli of C15-type LaCo₂. 

Additionally, the shear moduli and Young’s modulus are the measure of resistance to reversible deformation by the applied shear force. The value of G and E increases with increasing the external pressure. this implies that, for LaCo₂ the resistance to shear deformation and the stiffness are increased with the increase of pressure respectively.

Poisson’s ratio v is another rule used for determining the ductility and brittleness of a compound. The material has a ductile nature if its value is greater than 0.26; otherwise, the material is considered to be brittle. Table III shows that Poisson’s ratio for LaCo₂ is greater than 0.26, which confirms its ductile nature. Poisson’s ratio v also gives knowledge about the bonding nature in a material. According to this limit the type of material bonding will be ionic if ≈ 0 : 25, covalent if ≈ 0 : 10 and metallic if ≈ 0 : 33. For our compounds the v is found close to 0.33 and hence major bonding in these materials is metallic [30].

Zener’s anisotropy factor (A) reveals the degree of the elastic anisotropy of the solid. For a perfectly isotropic material, A will take the value equal to 1; deviation from 1 shows anisotropic nature of the material [³¹,³²]. For LaCo₂ the values of A which are listed in Table III, are calculated by using the following formula at different pressures:

The determined values of A are found to be different than unity, suggesting the material will have elastic anisotropic nature.

In addition, the melting temperature, which is another thermodynamic quantity, has been determined using the following empiric expression as mentioned below [³³,³⁴].

The corresponding value observed for the present alloy is 1400.69 ± 300 K thus shows that the material has a great capacity to preserve its crystal structure over a wide range of temperatures.

3.3 Electronic properties

Figure 5 displays the computed spin-polarized band structures for LaCo₂ at the measured equilibrium lattice constants along with the high symmetry directions in the Brillouin zone using the GGA+U method. This figure shows, in both spin configurations, an overlap between the valence and the conduction bands around the Fermi level. In addition, at the Fermi level there is no band gap, which indicates the metallic nature of our compound.

FIGURE 5 Band structures of LaCo₂ obtained within GGA + U for spin down and spin up configurations. 

It is well known that the density of states at E _F , N(E _F ) tends to uncommon transport properties, since both spin-up and spin-down densities contribute to the E _F states. Analogous behavior has been reported in TbFe₂, TbCo₂ and RENi₂ (RE= Dy, Ho and E _r ) compounds [³⁵,³⁶]. Consequently, the bands which cross E _F are responsible for the transport properties of the compound and the bands which do not cross E _F can make a negligent contribution to the transport properties [³⁵].

Furthermore, the total density of states of the compound (for all atoms in a unit cell) and the partial DOS (per atom) were determined using GGA + U approximation. The curve is shown in Fig. 6. The spin-up and spin-down, along with the TDOS, are found to be anti-symmetrical, which indicates that the LaCo₂ is in a ferromagnetic state. These findings can be due to the contribution of the 3d states of cobalt (Co) and the contribution of the 4f states of Lanthane (La) as well to the TDOS.

FIGURE 6 The total and partial density of states of LaCo₂ (spin up and spin down) obtained with GGA + U calculations. 

Moreover, the energy dispersion spectrum is observed to cross the Fermi level (E _F = 0 eV), which is another indicator that this compound is potentially electrically conductor. Indeed, the low energy region, which is located around −5 and 0 eV, is mainly filled by the d states of the Co atoms. The region above (E _F = 0) from 2 to 4 eV is dominated by 4f states of Lanthane (La) and the majority of the total DOS (from −5 to + 2 eV) is governed by the 3d states of Co.

3.4 Transport properties

The thermoelectric properties were performed within relaxation time approximation implemented in the BoltzTrap code, based on the Boltzmann transport theory. Effective thermoelectric materials should have high electrical conductivity (σ/τ), large Seebeck coefficient (S) and low thermal conductivity (k _e /τ). We need to maintain the density and mobility of the charge carriers in order to achieve maximum electrical conductivity.

3.4.1 Electrical conductivity

To ensure that the examined compound is sufficient to provide maximum effectiveness in the E region. We have plotted the electrical conductivity of LaCo₂ compound at three constant temperatures 300, 600 and 900 K as a function of chemical potential (µ − E _F = ±2 eV) for both spin polarized channels as shown in Fig. 7. From the figure, we can see that for spin up channel of LaCo₂ a remarkable rise in electrical conductivity occurs just above and below E _F to attain the maximum value of approximately 3.73 × 10²⁰ (Ωms)⁻¹ around µ−E _F = −0.11 eV at temperature 300 K for n-type conductions, and 3.78 × 10²⁰ (Ωms)⁻¹ at 300 K around µ−E _F = 0.09 eV for p-type conductions. While for the spin down case a maximum value of about 4.01 × 1020 (Ωms)⁻¹ was achieved at 300 K and µ − E _F = −0.08 eV for n-type conductions and about 2.96 × 1020 (Ωms)⁻¹ at 300 K and µ − E _F = −0.09 eV for p-type conductions.

FIGURE 7 The electrical conductivity of LaCo₂ for both spin configurations as a function of chemical potential at three constant temperatures (300, 600 and 900) K. 

It is obvious from the above interpretation that the holes play an major role in achieving the highest values of electrical conductivity in the spin up configuration and that the electrons play a important role in the spin-down configuration.

3.4.2 Electronic thermal conductivity

The thermal conductivity is generally composed of two parts, the first one coming from the electronic contribution κ _e (electron or hole) and the second one generated by lattice thermal conductivity κ _l (phonon contribution). We should underline that BoltzTraP code only calculates the electronic part κ _e .

The electronic thermal conductivity is measured As a function of chemical potential (µ−E _F = ±0.15 eV) for both spin configurations of LaCo₂ compound. Figure 8 shows that a significant increase in thermal conductivity occurs with an increase in temperature. And the temperature 900 K generated the highest κτ values, while 300 K is the adequate temperature giving the lowest κτ/e values within the chemical potential range µ−E _F = +0.15 eV. So the materials studied provided optimal efficiency in this range of chemical potential.

FIGURE 8 The electronic thermal conductivity of LaCo₂ for both spin configurations as a function of chemical potential at three constant temperatures (300, 600 and 900) K. 

3.4.3 Seebeck coefficient (Thermopower)

The Seebeck coefficient indicates the magnitude of an induced thermoelectric voltage regarding the temperature difference throughout the material [¹]. Positive Seebeck coefficient values signify that hole-type carriers dominate the thermoelectric transport in the material, while negative values of the Seebeck coefficient signify the dominance of electron type carriers.

Figure 9 illustrates the chemical potential (µ − E _F = ±0.2F eV) dependence of Seebeck coefficient of LaCo₂ compound at temperature values of 300 K, 600 K and 900 K for both spin configurations. It has been found that for spin up channel of LaCo₂ the highest value of Seebeck coefficient for p-type conductions of about 64.02 µV/K is achieved at 300 K for µ − E _F = 0.14 eV. And for n-type conductions of about −29.83 µV/K at 900 K for µ − E _F = −0.014 eV. While for spin down channels of the highest value of Seebeck coefficient for p-type conductions is about 45.74 µV/K at 300 K for µ−E _F = 0.14 eV, while for n-type conductions is about −47.09 µV/K at 600 K for µ − E _F = 0.017 eV.

FIGURE 9 The Seebeck coefficient of LaCo₂ for both spins configurations as a function of chemical potential at three constant temperatures (300, 600 and 900) K. 

3.4.4 Power factor (P)

The power factor (PF=S² σ), which is essential to explore the efficiency of the thermoelectric material, is plotted against chemical potential µ − E _F = ± E 0.15 eV in Fig. 10 at three temperatures 300, 600 and 900 K for both spin configuration of LaCo₂. In the spin up channel, the highest value of power factor of about 4.1 × 10¹⁴ W/mK²s is achieved at 900 K for µ − E _F = −0.07 eV, 2.42 × 10¹⁴ W/mK²s is achieved at 600 K for µ − E _F = −0.07 eV and 1.16 × 1014 W/mK² s is achieved at 300 K for µ − E _F = 0.14 eV. Although the spin down channel, shows the highest value of about 4.5×10¹⁴ W/mK²s at 900 K for µ−E _F = −0.1 eV and 2.67×10¹⁴ W/mK²s at 600 K for µ−E _F = −0.11 eV, while it is 0.92 × 10¹⁴ W/mK²s at 300 K for µ − E _F = −0.11 eV.

FIGURE 10 The power factor of LaCo₂ for both spin configurations as a function of chemical potential at three constant temperatures (300, 600 and 900) K.