Pressure dependent half-metallic ferromagnetism in inverse Heusler alloy Fe2CoAl: a DFT+U calculations

We report the electronic and magnetic properties along with the Curie temperature (TC) of the inverse full Heusler alloy (HA) Fe2CoAl obtained by using the first-principles computational method. Our calculations suggests that Fe2CoAl is a magnetic metal when treated within PBE-GGA under the applied compressive pressures. However, the implementation of electron–electron (U) (i.e., GGA+U) with varying compressive pressure (P) drastically changes the profile of the electronic structure. The application of GGA+U along with pressure induces ferromagnetic half-metallicity with an integer value of total magnetic moment ∼4.0 μB per unit cell. The integer value is in accordance with the Slater–Pauling's rule. Here, we demonstrate the variation of semiconducting gap in the spin down channel. The band gap increases from 0.0 eV to 0.72 eV when increasing the pressure from 0 to 30 GPa. Beyond 30 GPa, the electronic band gap decreases, and it is completely diminished at 60 GPa, exhibiting metallic behaviour. The analysis of the computed results shows that the treatment of electron–electron interactions within GGA+U and the application of compressive pressure in Fe2CoAl enables d–d orbital hybridization giving rise to a half-metal ferromagnet. The TC calculated from mean field approximation (MFA) decreases up to 30 GPa and then increases linearly up to 60 GPa.


Introduction
The discovery of the prototype Heusler alloy Cu 2 MnAl in 1903 (ref. 1) initiated new research interest due to its ferromagnetic behaviour despite having all non-magnetic constituents. 1, 2 Heusler alloys have been studied for 120 years, but the application of these materials in spintronics has only been considered for 30-40 years. [3][4][5][6][7][8][9][10][11][12] Half-metal ferromagnetism (HMF) is the top priority among all the explored properties of Heusler alloys. [13][14][15][16][17][18][19][20] HMF occurs when magnetic materials exhibit metallic behavior in one of the spin channels whereas the other spin channel is semiconducting. Rigorous research has been performed in order to understand the underlying mechanism of unusual ferromagnetic behaviour in terms of atomistic scale interactions. Several theories and models have been proposed for the integration of these exotic properties in new technological applications in spintronics. 21 Spintronics is a branch of science which deals with the charge and spin of an electron. High capacity storage devices, magnetic RAM, spin-injection, spin-valves, spin-lters, GMR, TMR, and many more potential electron spin-based components are foreseen. [22][23][24][25] To improve the storage capacity of memory devices, the manipulation of spin degrees of freedom is crucial. Fabrication of practical devices with enhanced efficiency is another challenging task. Some of the Heusler alloys with high spin polarization and high Curie temperature 21,26 are promising and may complement this goal. Although the practical application of HMF in spintronic devices is in the preliminary stage, lab work is still in progress. Heusler alloys with 3d-orbitals are very sensitive to externally applied elds (temperature, pressure, electric and magnetic elds), and as a result their electronic and magnetic properties can be tuned very readily. The d-d orbital hybridization between the transition metals is reported to be responsible for some of their outstanding electronic and magnetic properties, including the high value of spin polarization at the Fermi level (E F ). 18 They exhibit low Gilbert damping, high Curie temperature, high spin polarization, etc. [27][28][29][30][31][32][33][34][35][36][37] In addition, Heusler alloys are also explored as potential thermoelectric materials 38 due to the presence of non-toxic components and their large range of working temperatures as compared to other thermoelectric materials. [39][40][41][42] Currently, numerous theoretical and experimental research investigations are in progress to study the electronic and magnetic properties of Heusler alloys via structural modications under applied strain and hydrostatic pressure. Under applied hydrostatic pressure, the bond lengths vary due to the displacement of atoms from their mean positions and changes electronic charge densities which have a direct impact on the electronic and magnetic properties. 43,44 Ram et al. have reported signicant changes in the band structure of Co 2 XY (X ¼ Cr, Mn and Y ¼ Al, Ga) type direct full-Heusler alloys under applied pressure. 45 Rasul et al. 46 have studied the quaternary Heusler alloys ScNiCrX (X ¼ Al, Ga) using a rstprinciples approach and reported the robustness of the halfmetallicity up to 6 GPa and 16 GPa for ScNiCrAl and ScNiCrGa, respectively. Amudhavalli et al. 47 49 There are reports on the variation of T C of various Heusler alloys (such as Ni-Mn based alloys) under applied pressure. [50][51][52] Dannenberg et al. 53 reported the structural dependence of T C of Fe 2 CoGa: $780 K for L2 1 and $770 K for L1 0 , respectively. Some more promising results for T C s for Febased full-Heusler alloys 54 under ambient conditions are: 925 K (Fe 2 CoGe), 750 K (Fe 2 NiGe), 845 K (Fe 2 NiGa), 798 K (Fe 2 CuGa), and 875 K (Fe 2 CuAl). In our previous work 55 only Co 2 FeAl exhibited half-metallicity with the implementation of DFT+U whereas Fe 2 CoAl was metallic. However, there is a pseudo bandgap above the Fermi energy in the spin-down channel of Fe 2 CoAl within DFT+U. Thus we were encouraged to implement external pressure to observe the location of the bandgap. Interestingly, we have observed a projection of the bandgap in the Fermi level in the spin-down channel, giving rise to halfmetallicity. Therefore, we have applied pressure, which shortens the bond lengths and facilitates d-d hybridization, which may give the desired results. In this paper, we have also tried to explore the electronic and magnetic properties along with the T C under compressive pressure using the rstprinciples approach. We report the sensitiveness of the electronic and magnetic properties under lattice strain due to applied pressure.

Computational details
In terms of their chemical compositions, Heusler alloys are classied as binary (X 3 Z), ternary (X 2 YZ or XYZ), and quaternary (AXYZ) compounds, where A, X and Y are transition metals and Z is a p-block element. Binary HAs only have two distinguishable elements in the unit cell. In general, ternary HAs are of two types viz. full (X 2 YZ) and half (XYZ) HAs. The stoichiometric atomic ratios of full (X 2 YZ) and half (XYZ) ternary HAs are 2 : 1 : 1 and 1 : 1 : 1, respectively. On the other hand, quaternary HAs have an equiatomic stoichiometric ratio of 1 : 1 : 1 : 1. Herein, we focus on ternary full-HAs. Ternary full-HAs crystallize in two ordered phases: Cu 2 MnAl-type with space group Fm 3m (225) 35,56 and Hg 2 CuTi-type with the F 43m (216) space group. 57,58 The former is referred to as regular/direct (L2 1 ) and the latter is inverse/indirect with XA-structure. The atomic Wyckoff positions for both the XA-phase and the L2 1 -phase of Fe 2 CoAl are tabulated in Table 1. In the case of ternary inverse full-HAs the electronegativity of the X-atom is less than that of the Y-atom (X ¼ Fe and Y ¼ Co). Thus, our Fe 2 CoAl system is an inverse full-HA with the electronegativity of the Fe-atom being less than that of the Co-atom. The primitive and conventional unit cells of Fe 2 CoAl with XA-structure are shown in Fig. 1. To study the electronic and magnetic properties of the inverse full-Heusler alloy Fe 2 CoAl, we have performed rst-principles Density Functional Theory (DFT) calculations. 59,60 All electronelectron interactions were considered according to the generalized gradient approximation (GGA) within the Perdew-Burke-Ernzerhof (PBE) parametrization. 61 We are aware that GGA is inadequate for treating strongly correlated d-electrons to derive accurate electronic structures. On the other hand, the successful obtainment of accurate electronic properties by using GGA+U has already been reported for many Heusler alloys. 62-67 Therefore, we have applied a screened Coulomb interaction along with the conventional GGA (GGA+U or DFT+U) in order to deal with strongly correlated 3d electrons. 68 We have adopted U parameters of 3.82 eV for Fe and 3.89 eV for Co according to our previous work. 55,69 For computation, we have used the all electron orbital based full potential linearized augmented plane wave (FP-LAPW) basis set formalism as implemented in the WIEN2K package. 70 The non-spherical cut off value of angular momentum within the muffin tin (MT) sphere is l max ¼ 10. R MT Â K max ¼ 7 where K max is the maximum value of the reciprocal lattice vector in the plane wave expansion and R MT is the smallest muffin tin (MT) radius. To model the electronic structure, the rst Brillouin zone (BZ) was integrated by taking a 10 Â 10 Â 10 k-mesh grid within the Monkhorst package. The self-consistency convergence criterion was set to be 0.0001 Ry. The ferromagnetic-antiferromagnetic exchange energy J ij is a key parameter in determining the Curie temperature (T C ). A code based on the single-site coherent potential approximation within the Green's function approach called spin-polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) 71,72 was used for the calculation of T C .

Structural properties
Our rst-principles calculations start with the optimization of both the L2 1 (direct) and XA (inverse) structures with different magnetic congurations. The initial magnetic congurations set for our calculations are ferromagnetic Table 2). The variations in total energy versus the lattice constant a (Å) for both the direct and inverse phases with the FM, AFM1 and AFM2 magnetic congurations are shown in Fig. 2a and b. The XA-structure (inverse) with the FM conguration is energetically favourable with the minimum energy as shown by the blue line and the diamonds [see Fig. 2b]. The ground state energy and pressure as a function of volume are also presented in Fig. 2c 76 This result also agrees well with the results for several other analogous Fe-based inverse Heusler alloys whose lattice parameters vary from 5.5 to 6.2Å. 35,54,55,[77][78][79][80][81][82][83][84] Further, the cohesive energy has been calculated from eqn (1). The cohesive energy results are presented in Fig. 7c and conrm the ground state stability of each system under different applied pressures.
Here in eqn (1), E Fe1 , E Fe2 , E Co , E Al and E T are the individual energies of Fe1, Fe2, Co, and Al, and the total energy of the system, respectively. n ¼ 4 denotes the total number of atoms in the unit cell.

Electronic and magnetic properties
We have investigated the electronic properties of Fe 2 CoAl by calculating the total density of states (TDOS) and energy bands at different pressures using GGA+U [cf. Fig. 3-5]. We have already reported the inadequacy of GGA in deriving the electronic properties in our previous work. 55 On the other hand, GGA+U has predicted a band gap in the spin down channel but this is well above the Fermi level (E F ). We have observed the presence of some band edges at the E F , mostly originating from Fe1-d (t 2g ), Fe2-d (e g , t 2g ) and Co-d (e g , t 2g ) [cf. Fig. 4a-d].
However, the Fe1-d (e g ) state is hardly seen in the picture as it lies far below the E F in the spin up channel [cf. Fig. 4a]. Hence, we have proceeded with our calculations by using GGA+U along with the application of compressive pressure. Interestingly, on the application of compressive pressure (5 GPa), the E F is pushed upward within the band gap [ Fig. 3a and b]. The halfmetallic band gap is attributed to the d-d hybridization between Fe2-d and Co-d followed by Fe1-d states to give bonding-antibonding states as in the case of other full-Heusler alloys. [15][16][17][18][19]26 The bonding states at the top of the valence band   Fig. 7d]. This result can also be conrmed by the energy band structure which exhibits the indirect nature of the band gap as measured along the L-X symmetry and denoted by the green circles [cf. Fig. 5]. In Fig. 3 and 4, we observe a large band gap of $0.72 eV at 30 GPa with the E F pinned exactly in the middle of the band gap, providing more evidence of half-metallicity. A further increase in applied pressure (say beyond 30 GPa) decreases the band gap with driing of the upper band edge (CBM) towards lower energy (specically towards the E F ). This can be seen in Fig. 3b and 4ad in which the Fe1-d state has moved towards higher energy in the conduction band, taking part less in the d-d hybridization, and the coupled Fe2-d-Co-d bands are pushed towards lower energy (i.e. towards the E F ) in the spin down region. It looks like the E F shis from the lower to the upper edge of the halfmetallic band gap in the spin down channel on increasing the pressure, a typical feature of band ip. At 60 GPa the CBM (upper edge of the band gap) appears at the E F , thus diminishing the half-metallic behaviour and increasing the metallicity. The robustness of the half-metallicty is measured in terms of spin polarization at the E F . The degree of spin polarization in the vicinity of the E F can be analyzed by and N Y (E F ) are the numbers of states at E F for the spin up and spin down channels, respectively. Our GGA+U calculation with pressure has signicantly improved the spin polarization by more than 45%. At 5 GPa we estimated $98% spin polarization. On varying the pressure, 5 < P < 60 GPa, we have achieved perfect half-metallic behaviour in our Fe 2 CoAl system. The projection of E F inside the band gap in the spin down channel and the nite value of the electron density around E F in the spin up channel at 5 < P < 60 GPa results in 100% spin polarization. To analyse the nature of the bonding between the atoms we have calculated the valence electron charge density (eÅ À3 ) and the results at 30 GPa are presented in Fig. 6. The analysis of the charge density shows the presence of metallic bonding between the atoms and no sign of covalent bonds. As reported elsewhere, the magnetic properties of perfect half-metallic ferromagnet (HMF) Heusler alloys (HAs) can be  predicted from the total number of valence electrons present in the unit cell. [15][16][17][18]26 The total magnetic moment can be derived from the Slater-Pauling (SP) rule 18 as given by M t ¼ (Z t À 24) m B , where M t is the total magnetic moment and Z t is the total number of valence electrons. The total number of valence electrons in our Fe 2 CoAl system is 2 Â 8 + 9 + 3 ¼ 28. So the expected value of the total magnetic moment is M t ¼ 4.0 m B which results in the half-metallicity. The total magnetic moment obtained from GGA sharply deviates from the Slater-Pauling rule. However, on treating the system within GGA+U and applying pressures (0-70 GPa), the total magnetic moment M t varies around 4.0 m B . The highest M t ¼ 4.440 m B and the lowest M t ¼ 3.981 m B , calculated at 0 GPa and 70 GPa, respectively [ Table 3]. The M t values at 0 and 70 GPa do not comply with the SP rule. Also, at 60 GPa M t ¼ 3.988 m B and Dm B ¼ $0.012 or $0.3%. At applied pressures of 5 < P < 60 GPa, the M t value is $4 m B . Thus, we can claim that the M t values are in accordance with the SP rule 18 at 5 < P < 60 GPa. We also present the variation of the partial magnetic moments calculated with GGA and   Table 3 Total and partial magnetic moments (in m B ) calculated with GGA+U along with the Curie temperature T C (in K)  GGA+U under different pressures in Fig. 7a and their numerical values are tabulated in Table 3.
In order to further understand the magnetic interactions and magnetic properties, we have calculated the magnetic exchange energy by modeling the pair exchange interaction parameter J ij . The J ij parameter is computed using the Heisenberg model. 17 The exchange coupling interactions are mapped between the atoms sitting at the i and j sites separated by some distance as in eqn (2) is the atomic t-matrix of the magnetic impurities at site i for the spin up/down state, T ij [Y is the scattering path operator between the i and j sites for the spin up/down state, and Tr L is the trace over the orbital variables of the scattering matrices. The calculated exchange parameters J ij for a central Fe1 atom interacting with all other atoms (Al-Fe1, Fe1-Fe1, Fe2-Fe1 and Co-Fe1) as a function of R ij /a at different pressures are shown in Fig. 8a-d. The estimated J ij is a key parameter for obtaining the Curie temperature (T C ) in the mean-eld approximation (MFA) as given in eqn (3): where k B is the Boltzmann constant, c is the concentration of the impurities, and hs j i is the average j component of the unit vector s r j along the direction of magnetization. The J ij below R ij / a ¼ $1 (below 1) shows a nite stable value. Meanwhile, the J ij values above R ij /a ¼ $1 (above 1) either remain close to 0 meV or uctuate around 0 meV [cf. Fig. 8a-d]. The variation of the total J ij along with the calculated T MFA C as a function of R ij /a under different pressures are shown in Fig. 9a and b. We can see that on increasing the pressure up to 30 GPa the J ij parameter decreases systematically. This leads to a decrease in T MFA C . As shown in Fig. 8c the J ij value is governed by a strong interaction between Fe1 and Fe2. The lowest calculated value of T MFA C ¼ 1124.20 K at 30 GPa is mainly attributed to the low value of J ij due to the short range (R ij /a < $1) interaction between Fe1 and Fe2 [see inset (blue line) in Fig. 8c 30 GPa, whereas at high pressure, i.e., above 30 GPa they follow a similar trend. 49 The calculated T C values obtained from the mean eld approximation are tabulated in Table 3. The other method to estimate the T C in relation to the total magnetic moment (M t ) of HMF-HAs is given by eqn (4). 19,26 As we have already discussed the inefficiency of GGA in deriving the half-metallicity, estimating T C by taking the M t obtained from GGA is not justiable. Therefore, we have taken the M t values calculated with GGA+U and substituted in eqn (4) to obtain the T cal C . The estimated values of T cal C are presented in Table 3. The T cal C values vary from 826.640 K to 743.561 K on varying the pressure from 0 to 70 GPa. These results are in good agreement with the T C s of other analogous Fe-based inverse full-HAs. 54 We have noted that the T C obtained from eqn (4) appears to be independent of interaction strength as the variation of the total magnetic moment (M t ) is very small. Referring to Table 3, the T C values calculated with the MFA (eqn (3)) look much higher as compared to those calculated with eqn (4). The large values of T MFA C may arise due to inability to include the magnetic percolation effect within the mean eld approximation. 55

Conclusion
In summary, we have studied the structural, electronic and magnetic properties of the inverse HA Fe 2 CoAl along with the T C  values obtained from GGA and GGA+U under applied pressures. We have shown that the strong correlation mainly comes from the Fe-3d and Co-d states, and the inclusion of electron-electron interactions within GGA as the GGA+U formalism is essential to describe the electronic properties. The implementation of GGA+U along with compressive pressure (5 < P < 60 GPa) leads to half-metallic behavior with the opening of a spin minority band gap. The predicted integer value of the total magnetic moment in the inverse full-Heusler alloy Fe 2 -CoAl, $4.0 m B at 30 GPa, is in accordance with the Slater-Pauling rule, which supports the half-metallicity. The T C calculated from eqn (4) is 747 K at 30 GPa, in good agreement with the results of other Fe-based inverse full-HAs. However, the results from MFA are overestimated.

Conflicts of interest
There are no conicts to declare.