Introduction

Significant momentum in the study of intermetallic Heusler alloys has increased over the last decade as these systems exhibit numerous extraordinary capabilities in exposing the desired properties, extending from robust spin-polarization, half-metallic magnetism, magnetoresistance, shape memory effect, spin gapless semiconductor to giant magnetocaloric effect, phase transitions and thermoelectric effect1,2,3,4,5,6,7. The technological applications exploiting these properties have been achieved successfully. Spintronic and thermoelectric applications are the offshoots of half-metallic ferromagnetism (being castoff in spin injectors8, spin filters9, magnetic tunnel junctions10, spin valves11, random access memories12) and spin gapless attributes to the Seebeck effect useful for thermoelectric devices13. Within these dimensions, the materials with compatible lattice structure, high spin polarization and high Curie temperature are anticipated in practical spintronic applications. Magnetoelectronic devices mostly depend on the disproportionate number of majority and minority spin carriers, as exhibited ideally by half-metallic materials i.e. 100% spin polarization at the Fermi level. Such materials display the concoction properties of semiconductor and metal. Additional motive to delegate Heusler alloys in these applications is that these systems have the same crystallographic structure with different functional characteristics and some of them are even very close in electronic structure and composition14,15. Since the discovery of the NiMnSb Heusler alloy in 198316, a sequence of experimental as well as theoretical efforts (first principles simulations) were attempted to predict novel semiconductor or half-metallic systems. Among such compounds, transition metal based Heuslers have been widely investigated by material scientists worldwide. Predominantly, the Fe based Heusler structures constitute a vast family with semiconducting or half-metallic band profiles. For example, Fe2YSi (Y=Cr, Mn, Fe, Co, Ni) alloys were experimentally synthesized and predicted to be half-metallic alloys17. Fe2TiAl was reported to have thermoelectric applications18. Other materials like, Fe2TiSi, Fe2TiGe and Fe2ZrSi19, FeMnSi20, FeVRuSi21 and many more to report here have been investigated for their magnetic, semiconducting or mechanical properties. Using first-principle calculations, Fe2YZ (Y=V, Ti, Nb, Zr, Ta, Hf and Z=Al, Ga, In, Sn, Ge, Si) Heusler compounds with room temperature power factors 4 to 5 times larger than classical thermoelectrics were reported recently by Bilc et al.22. However, a little information is available on the electronic structure, mechanical stability, phonon dynamics and bonding characteristics of Fe2TaAl and Fe2TaGa alloys. In addition, the untouched lattice dynamical parameters and phonon properties are necessary to understand the intriguing physical properties and hence in this work, we tried to investigate their structural and mechanical stability, electronic and lattice dynamical properties in detail.

Results and Discussion

Structural and mechanical stability

Full-Heusler ternary alloys of Fe2TaAl and Fe2TaGa type have been found to crystallize in a cubic structure with space group (Fm-3m) as depicted in Fig. 122. The corresponding atomic locations are Fe (1/4, 1/4, 1/4), Ta (1/2, 1/2, 1/2) and Z (0, 0, 0). The ground state structure is determined by geometry optimization via the total energy per unit cell (see Supporting Information) and thereby the equilibrium lattice constants, derivative of bulk modulus, total energy and equilibrium volume are obtained as shown in Table 1. These optimized lattice parameters are fetched out to calculate the ground state properties of these alloys.

Figure 1
figure 1

Crystal structure of conventional unit cell for Fe2TaAl and Fe2TaGa in Fm-3m configuration.

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Table 1 Calculated values of the lattice constant (ao), unit cell volume (V0), derivative of bulk modulus (B’), ground-state energy (E0) and energy gaps (ΔE) of Fe2TaAl and Fe2TaGa alloys.
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The mechanical stability and elastic response of any crystal system is measured from the elastic constants and thereby the indication of its mechanical properties; e.g Bulk/Young’s/shear moduli, Debye or melting temperature, poisons ratio, brittle/ductile or hardness, etc are derived from them. Macroscopic distortion of a crystal structure is directly related to the elastic constants and is applicable in the evaluation of elastic energies or strains in materials under applied (internal/external/thermal) stresses23. These constants are determined by the linear response of a crystal towards the external forces, and are also associated with structural stability, equation of state (EOS), interatomic potential and phonon spectra24. Beyond this, the thermal properties including specific heat, thermal expansion, Debye temperature and Gruneisen parameter are exclusively linked to elastic constants. Consequently, the determination of elastic constants is indispensable to characterize a solid crystal. In this work, we have calculated the elastic constants of Fe2TaAl and Fe2TaGa systems using the generalized gradient approximation (GGA) and are listed in Table 2. For a simple cubic system, the elastic stiffness constants (Cij) are reduced to only three independent constants viz. C11, C12, and C44. From Table 2, it can be seen that the comprehensive Born criteria; C12 < B < C11, (C11 − C12) > 0, (C11 + 2C12) > 0 and C44 > 0 is rigorously followed by the observed lattice constants and hence their mechanical stability is confirmed25. Further, to get the information about the different elastic moduli and other derivables, Viogt-Reuss-Hill method is used via the Eqns (1–3)26,27,28,

$${G}_{V}=\frac{({C}_{11}-{C}_{12}+3{C}_{44})}{5}\,;\,{G}_{R}=\frac{5({C}_{11}-{C}_{12}){C}_{44}}{4{C}_{44}+3({C}_{11}-{C}_{12})}\,;\,G=\frac{{G}_{V}+{G}_{R}}{2}$$
(1)

where, bulk modulus is represented as,

$${B}_{V}={B}_{G}=B=\frac{({C}_{11}+2{C}_{12})}{3}$$
(2)

and, the Youngs modulus, Poisson’s ratio and anisotropy parameter are defined as

$${\rm{Y}}=\frac{9{\rm{BG}}}{3{\rm{B}}+{\rm{G}}}\,;\,\upsilon =\frac{3{\rm{B}}-{\rm{Y}}}{6{\rm{B}}}\,;\,A=\frac{2{{\rm{C}}}_{44}}{({{\rm{C}}}_{11}-{{\rm{C}}}_{12})}$$
(3)
Table 2 Calculated values of elastic (C11, C12, C44), bulk (B), Shear (G), Young’s (Y) moduli (in GPa), Poisson’s ratio (υ), Zener anisotropy factor (A), B/G ratio, Cauchy’s pressure (C”), and Melting Temperature (Tm) in K for Fe2TaAl and Fe2TaGa alloys.
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Calculated results for the values of elastic parameters (B, G, Y, A, v, C”, B/G, etc.) are summed up in Table 2. The hardness of a material is generally delivered from the observed values of bulk modulus (B) and shear modulus (G). Indisputably, when a stress is applied to any system, the opposition offered at the critical point before which the system is fractured refers to the hardness of that material29. Concurrently, the ductile and brittle character of materials is directly linked to the critical value (1.75) of B/G ratio. If B/G is less than the critical value, then the material is ductile otherwise it is said to be brittle30. The covalent materials (e.g. Diamond) in principle are relatively hard and obviously brittle with a smaller Pugh ratio. The strong covalent bonds in such materials certainly produce a significant resistance resulting in a quite high hardness. Conversely, ductile materials with a high Pugh’s ratio are characterized by metallic bonding and low hardness. Consequently, the observed values of B, G and B/G clearly determine the Fe2TaAl alloy to be much harder and ductile from Fe2TaGa compound. This claim is also supported by the (C” = C12–C44) values because its positive value defines the ductile nature of the present alloys and if its value is negative, then the system is said to be brittle in nature31. Positive Cauchy pressure also specify the presence of metallic bonding in a material while as the negative value demonstrates the directional (covalent) and angular bonds. In the present set of calculations, Cauchy pressure predicted for both the Heusler alloys are positive which reflects their metallic character. Poisson’s ratio and its critical value; 0 < υ < 0.5 simply defines the plasticity of a crystal. Its small value reflects the maximum plastic character and conversely, the material is elastic in nature32. So, the elastic nature of these alloys can be observed from the υ values mentioned in Table 2. At the same time, the value of anisotropic factor (A) for a perfectly isotropic system is equal to 1 and its value below or above unity proposes the anisotropic character of a compound. Therefore, the Fe2TaAl alloy is purely isotropic but the Fe2TaGa alloy is anisotropic in nature31.

The thermodynamic behavior of the Fe-based Heusler systems has been described by the calculation of melting temperature (Tm). Using the empirical equation (Eqn. 4) given below, the melting temperature of the present Heusler systems has been premeditated33,34.

$${T}_{m}(K)=[553(K)+(5.911){c}_{11}]GPa\pm 300K$$
(4)

The large values of Tm as seen from the Table 2 imply the strength of the present materials against the temperature and this hints about the retention of their ground state crystal structures at raised temperatures.

Semiconducting gap and electronic structure

Since both these alloys have 24 valence electrons (Zt) in their equilibrium structures, therefore, Slater-Pauling rule34; (Mt = Zt − 24) comprises the zero-spin magnetic moment for both these materials. Their non-magnetic character has earlier been reported in ref.22. The groundwork of structural optimization yields the equilibrium lattice constant, and the same is utilized to calculate the electronic band profile of Fe2TaAl and Fe2TaGa Heusler materials. Three different schemes; generalized gradient approximation (GGA), onsite Hubbard approximation (GGA + U) and modified Beckhe Johnson (mBJ) schemes have been employed and the spin polarized band profiles are displayed in Figs 2 and 3. The GGA and mBJ calculated energy gaps are very small as compared to GGA + U calculations. However, GGA + U clearly widens the gap between valence and conduction bands in both the compounds. From GGA + U calculations, Fe2TaAl is observed to be a direct band gap material with Eg = 1.80 eV, while as Fe2TaGa is observed to be indirect band gap semiconductor with the corresponding energy gap of 1.30 eV. In the latter case, the valence band maximum (VBM) occurs at the Γ symmetry point and the conduction band minimum (CBM) occurs at the X symmetry point in its Brillouin zone. However, GGA and mBJ methods reveal both these materials to be the p-type indirect band gap semiconductors.

Figure 2
figure 2

Calculated band structures of Fe2TaAl by GGA, mBJ and GGA + U methods.

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Figure 3
figure 3

Calculated band structures of Fe2TaGa by GGA, mBJ and GGA + U methods.

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Further, the spin polarized total density of states (TDOS) depicted in Fig. 4 and partial density of states (PDOS) shown in Fig. 5 for the present materials at their equilibrium lattice constants are discussed. Since, the total DOS in up and down spin channels are same (cancel each other), therefore the non-magnetic character of these alloys can be estimated. Also, the pDOS of these materials by GGA underestimates the band structure and therefore small or negligible gaps are observed by this method. Consequently, the onsite Hubbard correction (described in section 4 for Fe-d states) widens the band gap comparatively. The Fe-d and Ta-d states are mostly populated around the Fermi level with a maximum contribution towards the total DOS and consequently, the corresponding bonding-antibonding states control the energy gap formation. At the same time group IV atomic states are less active around the Fermi level in these materials. Thus, the observed band gap in these alloys is due to the typical d-d hybridization between the valence states of Fe and Ta atoms and the same has been explained elsewhere for similar materials like Co2TaAl and Co2TaGa3,28. Hence, from the observed band profiles and densities of state plots, both the compounds are found to be p-type indirect band gap semiconductors. Present calculations unlock the potential application of these alloys in semiconductor and energy harvesting technologies.

Figure 4
figure 4

Observed total densities of states (DOS) and partial densities of states (pDOS) of Fe2TaAl compound calculated by GGA, mBJ and GGA + U schemes.

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Figure 5
figure 5

Observed total densities of states (DOS) and partial densities of states (pDOS) of Fe2TaGa compound calculated by GGA, mBJ and GGA + U schemes.

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Phonon properties and Cohesive Energies

Phonon dispersions are interesting phenomenon to understand the dynamical stability of a crystal system above and beyond knowing their thermal behavior, superconductivity, Raman and thermal spectroscopy35,36,37. Figure 6(a,b) shows the phonon dispersion curves (PDCs) of these alloys obtained using GGA scheme. The four atoms of Fe2TaAl and Fe2TaGa system in its unit cell give rise to three acoustical and nine optical modes constituting a total of 12 phonon branches. But, in different directions, this number is reduced due to degeneracy. The optical modes obtained for Fe2TaAl are at around 510.19 cm−1 and 680.22 cm−1 and for Fe2TaGa are around 470.29 cm−1, 590.63 cm−1 at zone center Г and W points.

Figure 6
figure 6

Phonon dispersion for Fe2TaAl and Fe2TaGa alloys.

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It can also be seen from Fig. 4(a,b) that the longitudinal optical-transverse optical (LO-TO) splitting in our materials is almost nil at Г and W symmetry points. The difference of LO and TO (ωLO − ωTO) is called a reststrahlen band, which estimates the number of reflected electromagnetic waves. In our calculation \({\omega }_{LO}^{2}-{\omega }_{TO}^{2}\approx 0\) at Г and W points, hence exhibit higher phonon scattering ability. Such phenomenon can obviously enhance the thermoelectric response of these materials by decreasing thermal conductivity. The dynamical stability is confirmed by the absence of imaginary frequency in all high symmetry direction for both the investigated compounds.

Since, the stability of both these materials has been confirmed via Convex Hull analysis with −303.3 MeV for Fe2TaAl and −64.3 MeV for Fe2TaGa as the energy above the hull which is equal to the energy of formation of the compounds of interest from the phases that would be stable if the compound did not exist22. Therefore, our phonon results strengthen the stability criteria of these alloys. Moreover, we have calculated the cohesive energies (Ecoh) of these compounds using the following equation:

$${E}_{Coh}=(2{E}_{Fe}^{iso}+{E}_{Ta}^{iso}+{E}_{Al/Ga}^{iso})-{E}_{F{e}_{2}TaAl/Ga}^{Total}$$

where \({E}_{Fe}^{iso}\), \({E}_{Ta}^{iso}\) and \({E}_{Al/Ga}^{iso}\) are the isolated atomic energies of the Fe, Ta and Al/Ga atoms, respectively, and \({E}_{F{e}_{2}TaAl/Ga}^{Total}\), is the total energy of Fe2TaAl and Fe2TaGa per formula unit. The measurement of strength of the binding force between the constituent atoms in a solid structure is confirmed from cohesive energy its positive value indicates the stability of the material. The calculated values of the cohesive energy are 22.71 eV for Fe2TaAl compound and 20.67 eV for Fe2TaGa alloy. These observed values are significant and comparable to Co2TaSi (21.76 eV), Co2TaGe (19.18 eV)38 and Hf2Val (21.56 eV)39; and this specifies the large value of chemical bond energy. Therefore, the stability of the present materials is confirmed from the mechanical stability criteria, cohesive energy and phonon dispersion results.

Conclusion

In this study, the electronic and mechanical properties of cubic Fe2TaAl and Fe2TaGa alloys have been examined using the density functional theory. Calculations of the electronic structure shows the indirect band gaps along Γ-L symmetry for all these materials. These compounds directly fall under the Slater Pauling rule that depicts their semiconducting behavior. Thus, we have confirmed from our studies that these full-Heusler alloys are narrow band gap semiconductors with 0.80 eV for Fe2TaAl and 0.61 eV for Fe2TaGa. In relation to electronic structure, the mechanical properties have been calculated. A theoretical study predicts the hardness and ductile nature of these materials and their ductility increases from Al > Ga and are potentially the possible hard semiconductor materials. In this study, the electronic, phonon and mechanical properties of cubic Fe2TaAl and Fe2TaGa alloys have been examined using the density functional theory. The calculation of lattice vibrations from DFPT gives the phonon dispersion. The coupling of optical and acoustic mode is an interesting phenomenon that we have observed in this study with zero LO-TO splitting at Г and W points of symmetry. The zero value of reststrahlen band is responsible for higher value of phonon scattering and reduced the lattice thermal conductivity.

Methodology

The investigation of structural, elastic, electronic and phonon properties of the Fe2TaAl and Fe2TaGa compounds, first-principles calculations using the full potential linearized augmented plane-wave method (FPLAPW)40 as executed in the WIEN2k Package41. This quantum mechanical code can precisely simulate the ground state structure, band gaps, dielectric properties, magnetic properties, and so on. The generalized gradient approximation (GGA)42, mBJ43 and onsite Hubbard correction (GGA + U)44, were adopted for the exchange-correlation potentials. In GGA + U calculations, we have used the U-J = 4.0 eV for Fe-d electrons with J = 0 according to the Dudarav’s method45. A dense k-point mesh of 10 × 10 × 10 was used in the Brillouin zone integrations. The basis functions are expanded up to RMTKmax = 7, where RMT is the smallest atomic radius in the unit cell and Kmax refers to the magnitude of the largest k vector in the plane wave expansion. The total energy convergence tolerance for the calculations was selected within 1 × 10−6 eV. Furthermore, the elastic properties were calculated by cubic elastic code46 and from these elastic constants other mechanical parameters were determined to discuss the mechanical response of these materials. DFT has proven to be one of the successful methods to predict the stability as well as ground state properties of the materials theoretically47,48. The lattice dynamical properties were analyzed in term of phonon frequencies as a second-order derivative of the total energy with respect to atomic displacements within the frame work of DFPT49 by using the Quantum Espresso package50.