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

D. Rai, L Lalrinkima, L Lalhriatzuala, L. Fomin, I. Malikov, Adlane Sayede, Madhav Prasad Ghimire, R. Thapa, Lalthakimi Zadeng

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D. Rai, L Lalrinkima, L Lalhriatzuala, L. Fomin, I. Malikov, et al., Pressure dependent half-metallic ferromagnetism in inverse Heusler alloy Fe2CoAl: a DFT+U calculations. RSC Advances, 2020, 10 (73), pp.44633-44640. 10.1039/d0ra07543d. hal-03205843

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Cite this: RSC Adv., 2020, 10, 44633

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

D. P. Rai, ID *a Lalrinkima,ab Lalhriatzuala,ª L. A. Fomin, ID c I. V. Malikov, c Adlane Sayede, ID d Madhav Prasad Ghimire, ID *e R. K. Thapa b and Lalthakimi Zadeng ID b

We report the electronic and magnetic properties along with the Curie temperature ($T_C$) 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 $\mu_B$ 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 $T_C$ calculated from mean field approximation (MFA) decreases up to 30 GPa and then increases linearly up to 60 GPa.

Received 2nd September 2020

Accepted 5th November 2020

DOI: 10.1039/d0ra07543d

rsc.li/rsc-advances

## 1 Introduction

The discovery of the prototype Heusler alloy Cu₂MnAl in 1903 (ref. 1) initiated new research interest due to its ferromagnetic behaviour despite having all non-magnetic constituents.<fnref n="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.³⁻¹² Half-metal ferromagnetism (HMF) is the top priority among all the explored properties of Heusler alloys.¹³⁻²⁰ 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-filters, GMR, TMR, and many more potential electron spin-based components are foreseen.²²⁻²⁵ 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 temperature²¹,²⁶ 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 fields (temperature, pressure, electric and magnetic fields), 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.²⁷⁻³⁷ In addition, Heusler alloys are also explored as potential thermoelectric materials³⁸ due to the presence of non-toxic components and their large range of working temperatures as compared to other thermoelectric

<footnote>*Physical Sciences Research Center, Department of Physics, Pachhunga University College, Aizawl 796001, India. E-mail: dibya@pucollege.edu.in</footnote>

<footnote>ᵇDepartment of Physics, Mizoram University, Aizawl 796009, India</footnote>

<footnote n="1">¹Institute of Microelectronics Technology and High Purity Materials RAS, 142432 Chernogolovka, Russia</footnote>

<footnote n="4">⁴Univ. Artois, CNRS, Centrale Lille, ENSCL, Univ. Lille, UMR 8181-UCCS-Unité de Catalyse et Chimie du Solide, F-62300 Lens, France</footnote>

<footnote n="5">⁵Central Department of Physics, Tribhuvan University, Kathmandu, Nepal. E-mail: madhav.ghimire@cdp.tu.edu.np</footnote>
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materials.39–42 Currently, numerous theoretical and experimental research investigations are in progress to study the electronic and magnetic properties of Heusler alloys via structural modifications 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 significant changes in the band structure of Co₂XY (X = Cr, Mn and Y = Al, Ga) type direct full-Heusler alloys under applied pressure.45 Rasul et al.46 have studied the quaternary Heusler alloys ScNiCrX (X = Al, Ga) using a first-principles approach and reported the robustness of the half-metallicity up to 6 GPa and 16 GPa for ScNiCrAl and ScNiCrGa, respectively. Amudhavalli et al.47 studied Fe-based ferromagnetic quaternary Heusler alloys and reported a pressure induced structural phase transition at 151.6 GPa, 33.7 GPa, 76.4 GPa, 85.3 GPa, 87.7 GPa and 96.5 GPa for CoFeTiSi, CoFeTiGe, CoFeTiAs, NiFeTiSi, NiFeTiGe and NiFeTiAs, respectively. The same group,48 performing similar kinds of studies, have reported a half-metal to metal phase transition for Co₂TiAl, Co₂TiGa and Co₂TiIn under applied external pressures of 76.5 GPa, 73.1 GPa and 63.9 GPa, respectively. Rambabu et al. studied the variation of the Curie temperature (T<sub>C</sub>) under applied pressure and reported enhanced T<sub>C</sub> at high compressive pressures.49 There are reports on the variation of T<sub>C</sub> of various Heusler alloys (such as Ni-Mn based alloys) under applied pressure.50–52 Dannenberg et al.53 reported the structural dependence of T<sub>C</sub> of Fe₂CoGa: ~780 K for L<sub>2</sub>1 and ~770 K for L<sub>1</sub>0, respectively. Some more promising results for T<sub>C</sub>S for Fe-based full-Heusler alloys54 under ambient conditions are: 925 K (Fe₂CoGe), 750 K (Fe₂NiGe), 845 K (Fe₂NiGa), 798 K (Fe₂CuGa), and 875 K (Fe₂CuAl). In our previous work55 only Co₂FeAl exhibited half-metallicity with the implementation of DFT+U whereas Fe₂CoAl was metallic. However, there is a pseudo bandgap above the Fermi energy in the spin-down channel of Fe₂CoAl 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 half-metallicity. 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<sub>C</sub> under compressive pressure using the first-principles approach. We report the sensitiveness of the electronic and magnetic properties under lattice strain due to applied pressure.

## 2 Computational details

In terms of their chemical compositions, Heusler alloys are classified as binary (X₂Z), ternary (X₂YZ 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 (X₂YZ) and half (XYZ) HAs. The stoichiometric

atomic ratios of full (X₂YZ) 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 crystalize in two ordered phases: Cu₂MnAl-type with space group Fm$\bar{3}$m (225)35,56 and Hg₂CuTi-type with the F$\bar{4}$3m (216) space group.57,58 The former is referred to as regular/direct (L<sub>2</sub>1) and the latter is inverse/indirect with XA-structure. The atomic Wyckoff positions for both the XA-phase and the L<sub>2</sub>1-phase of Fe₂CoAl 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 = Fe and Y = Co). Thus, our Fe₂CoAl 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 Fe₂CoAl with XA-structure are shown in Fig. 1. To study the electronic and magnetic properties of the inverse full-Heusler alloy Fe₂CoAl, we have performed first-principles Density Functional Theory (DFT) calculations.59,60 All electron-electron 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

<table><caption>Table 1 Wyckoff positions of the Fe₂-based inverse full-Heusler alloy Fe₂CoAl</caption><thead><tr><th>Positions</th><th>x</th><th>y</th><th>z</th></tr></thead><tbody><tr><td><strong>Inverse</strong></td><td></td><td></td><td></td></tr><tr><td>Fe1</td><td>0.75</td><td>0.75</td><td>0.75</td></tr><tr><td>Fe2</td><td>0.50</td><td>0.50</td><td>0.50</td></tr><tr><td>Co</td><td>0.25</td><td>0.25</td><td>0.25</td></tr><tr><td>Al</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td><strong>Direct</strong></td><td></td><td></td><td></td></tr><tr><td>Fe1</td><td>0.25</td><td>0.25</td><td>0.25</td></tr><tr><td>Fe2</td><td>0.75</td><td>0.75</td><td>0.75</td></tr><tr><td>Co</td><td>0.50</td><td>0.50</td><td>0.50</td></tr><tr><td>Al</td><td>0.00</td><td>0.00</td><td>0.00</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Fig. 1 (a) Primitive and (b) conventional unit cell of the inverse Heusler alloy Fe₂CoAl with space group F$\bar{4}$3m. Balls in red, orange, blue and yellow in the represents Fe1, Fe2, Co and Al respectively.</figcaption></figure>
<figure><img src="image_5.png" /></figure>

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*<sub>max</sub> = 10. *R*<sub>MT</sub> × *K*<sub>max</sub> = 7 where *K*<sub>max</sub> is the maximum value of the reciprocal lattice vector in the plane wave expansion and *R*<sub>MT</sub> is the smallest muffin tin (MT) radius. To model the electronic structure, the first Brillouin zone (BZ) was integrated by taking a 10 × 10 × 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*<sub>ij</sub> is a key parameter in determining the Curie temperature (*T*<sub>C</sub>). 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*<sub>C</sub>.

# 3 Results and discussion

## 3.1 Structural properties

Our first-principles calculations start with the optimization of both the L2₁ (direct) and XA (inverse) structures with different magnetic configurations. The initial magnetic configurations set for our calculations are ferromagnetic FM (Fe1↑, Fe2↑, Co↑), antiferromagnetic AFM1 (Fe1↑, Fe2↓, Co↑), and AFM2 (Fe1↑, Fe2↑, Co↓) (see 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 configurations are shown in Fig. 2a and b. The XA-structure (inverse) with the FM configuration 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 and d. The calculated lattice constant is 5.73 Å and is consistent with previously reported values of 5.70 Å,⁷³ 5.71 Å,⁷⁴ 5.766 ± 0.05 Å,⁷⁵ and 5.732 Å.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 6.2 Å.³⁵,⁵⁴,⁵⁵,⁷⁷⁻⁸⁴ Further, the cohesive energy has been calculated from eqn (1). The cohesive energy results are

<table><caption>**Table 2** Magnetic configuration (MC) of the individual atoms Fe1, Fe2, Co, and Al, and total ground state energy *E*<sub>T</sub> in Ry</caption><thead><tr><th>MC</th><th>Fe1</th><th>Fe2</th><th>Co</th><th>Al</th><th><i>E</i><sub>T</sub></th></tr></thead><tbody><tr><td><strong>Inverse</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FM</td><td>↑</td><td>↑</td><td>↑</td><td>N</td><td>-1074.852</td></tr><tr><td>AFM1</td><td>↑</td><td>↓</td><td>↑</td><td>N</td><td>-1074.829</td></tr><tr><td>AFM2</td><td>↑</td><td>↑</td><td>↓</td><td>N</td><td>-1074.8509</td></tr><tr><td><strong>Direct</strong></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FM</td><td>↑</td><td>↑</td><td>↑</td><td>N</td><td>-1074.818</td></tr><tr><td>AFM1</td><td>↑</td><td>↓</td><td>↑</td><td>N</td><td>-1074.792</td></tr><tr><td>AFM2</td><td>↑</td><td>↑</td><td>↓</td><td>N</td><td>-1074.758</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 2** Variation of total energy as a function of lattice constant *a* (Å): (a) direct phase, (b) inverse phase (AMF1, AFM2 and FM configurations are represented by black, red and blue lines, respectively), (c) ground state energy as a function of volume in Å³ for FM and (d) pressure in GPa as a function of volume in Å³.</figcaption></figure>

presented in Fig. 7c and confirm the ground state stability of each system under different applied pressures.

$$E_C = \frac{E_{Fe1} + E_{Fe2} + E_{Co} + E_{Al} - E_T}{4} \quad (1)$$

Here in eqn (1), *E*<sub>Fe1</sub>, *E*<sub>Fe2</sub>, *E*<sub>Co</sub>, *E*<sub>Al</sub> and *E*<sub>T</sub> are the individual energies of Fe1, Fe2, Co, and Al, and the total energy of the system, respectively. *n* = 4 denotes the total number of atoms in the unit cell.

## 3.2 Electronic and magnetic properties

We have investigated the electronic properties of Fe₂CoAl 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*<sub>F</sub>). We have observed the presence of some band edges at the *E*<sub>F</sub>, mostly originating from Fe1-d (t<sub>2g</sub>), Fe2-d (e<sub>g</sub>, t<sub>2g</sub>) and Co-d (e<sub>g</sub>, t<sub>2g</sub>) [cf. Fig. 4a–d]. However, the Fe1-d (e<sub>g</sub>) state is hardly seen in the picture as it lies far below the *E*<sub>F</sub> 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*<sub>F</sub> is pushed upward within the band gap [Fig. 3a and b]. The half-metallic 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.¹⁵⁻¹⁹,²⁶ The bonding states at the top of the valence band
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 3 (a) Total DOS calculated using GGA+U, (b) partial DOS of Fe1-d, Fe2-d, Co-d and Al-p calculated with GGA+U at different pressures.</figcaption></figure>

form the valence band maximum (VBM) and the lowest anti-bonding states in the conduction region form the conduction band minimum (CBM). On the other hand, the spin up channel is still conducting. This hybrid characteristic of being semi-conducting in the spin down channel and conducting in the spin up channel results in peculiar half-metal ferromagnetic (HMF) behaviour. We have also noticed widening of the band gap with increasing compressive pressure, as this facilitates hybridization due to shortening of the bond lengths. The energy band gap increases from 0.0 eV to 0.72 eV on increasing the pressure from 0 to 30 GPa [cf. Fig. 7d]. This result can also be confirmed 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 drifting of the upper band edge (CBM) towards lower energy (specifically towards the $E_F$). This can be seen in Fig. 3b and 4a-d 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 $E_F$ shifts from the lower to the upper edge of the half-metallic band gap in the spin down channel on increasing the pressure, a typical feature of band flip. 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-metallicity 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 $P = [N_↑(E_F) - N_↓(E_F)]/[N_↑(E_F) + N_↓(E_F)]$, where $N_↑(E_F)$ and $N_↓(E_F)$ are the numbers of states at $E_F$ for the spin up and spin down channels, respectively. Our GGA+U calculation with pressure has significantly 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 Fe₂CoAl system. The projection of $E_F$ inside the band gap in the spin down channel and the finite 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 ($e Å⁻³$) 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

<figure><img src="image_9.png" /><figcaption>Fig. 4 (a) Partial DOS of Fe1 (d-e<sub>g</sub>, d-t<sub>2g</sub>), (b) partial DOS of Fe2 (d-e<sub>g</sub>, d-t<sub>2g</sub>), (c) partial DOS of Co (d-e<sub>g</sub>, d-t<sub>2g</sub>) and (d) partial DOS of Al-p at different pressures.</figcaption></figure>
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 5 Band energies of Fe₂CoAl calculated with GGA+U at (a) *P* = 0 GPa, (b) *P* = 5 GPa, (c) *P* = 10 GPa, (d) *P* = 20 GPa and (e) *P* = 60 GPa.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 6 Electron density (e Å⁻³) of Fe₂CoAl along the (a) 001, (b) 011 and (c) 111 directions.</figcaption></figure>

predicted from the total number of valence electrons present in the unit cell.¹⁵⁻¹⁸,²⁶ The total magnetic moment can be derived from the Slater-Pauling (SP) rule¹⁸ as given by $M_t = (Z_t - 24) \mu_B$, 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 Fe₂CoAl system is $2 \times 8 + 9 + 3 = 28$. So the expected value of the total magnetic moment is $M_t = 4.0 \mu_B$ 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 $4.0 \mu_B$. The highest $M_t = 4.440 \mu_B$ and the lowest $M_t = 3.981 \mu_B$, calculated at 0 GPa and 70 GPa, respectively [Table 3]. The $M_t$ values at 0 and 70 GPa do not comply

<table><caption>Table 3 Total and partial magnetic moments (in $\mu_B$) calculated with GGA+U along with the Curie temperature $T_C$ (in K)</caption><thead><tr><th>P (GPa)</th><th>M<sub>Tot</sub></th><th>M<sub>Fe1</sub></th><th>M<sub>Fe2</sub></th><th>M<sub>Co</sub></th><th>T<sup>cal</sup><sub>C</sub></th><th>T<sup>MFA</sup><sub>C</sub></th></tr></thead><tbody><tr><td>0.0</td><td>4.440</td><td>2.126</td><td>1.011</td><td>1.109</td><td>826.640</td><td>1164.30</td></tr><tr><td>5.0</td><td>4.030</td><td>2.158</td><td>0.989</td><td>1.075</td><td>752.430</td><td>1134.90</td></tr><tr><td>10.0</td><td>3.999</td><td>2.171</td><td>0.990</td><td>1.051</td><td>746.819</td><td>1130.60</td></tr><tr><td>20.0</td><td>3.990</td><td>2.165</td><td>1.010</td><td>1.018</td><td>745.190</td><td>1127.20</td></tr><tr><td>30.0</td><td>4.000</td><td>2.155</td><td>1.016</td><td>0.995</td><td>747.000</td><td>1124.20</td></tr><tr><td>40.0</td><td>3.996</td><td>2.152</td><td>1.009</td><td>0.979</td><td>744.828</td><td>1146.30</td></tr><tr><td>50.0</td><td>4.009</td><td>2.137</td><td>1.025</td><td>0.978</td><td>748.629</td><td>1173.50</td></tr><tr><td>60.0</td><td>3.988</td><td>2.121</td><td>1.028</td><td>0.964</td><td>746.276</td><td>1190.40</td></tr><tr><td>70.0</td><td>3.981</td><td>2.105</td><td>1.024</td><td>0.957</td><td>743.561</td><td>1187.00</td></tr></tbody></table>

with the SP rule. Also, at 60 GPa $M_t = 3.988 \mu_B$ and $\Delta\mu_B = \sim 0.012$ or $\sim 0.3\%$. At applied pressures of $5 < P < 60$ GPa, the $M_t$ value is $\sim 4 \mu_B$. Thus, we can claim that the $M_t$ values are in accordance with the SP rule¹⁸ at $5 < P < 60$ GPa. We also present the variation of the partial magnetic moments calculated with GGA and

<figure><img src="image_13.png" /><figcaption>Fig. 7 (a and b) Total and partial magnetic moments calculated with the FPLAPW method using WIEN2K, variation of (c) cohesive energy and (d) energy band gap as a function of pressure.</figcaption></figure>
<figure><img src="image_14.png" /></figure>

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)

$$J_{ij} = \frac{1}{4\pi} \int_{E_F} d(E) \text{Tr}_L \{ \Delta_i T_{\uparrow}^{ij} \Delta_j T_{\uparrow}^{ji} \}, \quad (2)$$

where $\Delta_i = t_{i\uparrow}^{-1} - t_{i\downarrow}^{-1}$, $t_{i\uparrow}^{-1}$, $t_{i\downarrow}^{-1}$ is the atomic t-matrix of the magnetic impurities at site i for the spin up/down state, $T_{\uparrow\downarrow}^{ij}$ 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-field approximation (MFA) as given in eqn (3):

$$k_B T_C^{\text{MFA}} \langle s_i \rangle = \frac{2}{3} c \sum_{j,r \neq 0} J_{ij}^{0,r} \langle s_j \rangle, \quad (3)$$

where $k_B$ is the Boltzmann constant, c is the concentration of the impurities, and $\langle s_j \rangle$ is the average j component of the unit vector $s_j^j$ along the direction of magnetization. The $J_{ij}$ below $R_{ij}/a \sim 1$ (below 1) shows a finite stable value. Meanwhile, the $J_{ij}$ values above $R_{ij}/a \sim 1$ (above 1) either remain close to 0 meV or fluctuate around 0 meV [cf. Fig. 8a-d]. The variation of the total $J_{ij}$ along with the calculated $T_C^{\text{MFA}}$ as a function of $R_{ij}/a$ under

<figure><img src="image_15.png" /><figcaption>Fig. 9 (a) Exchange interaction $J_{ij}$ (meV) and (b) Curie temperature $T_C^{\text{MFA}}$ (K) as a function of pressure.</figcaption></figure>

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 $T_C^{\text{MFA}}$. As shown in Fig. 8c the $J_{ij}$ value is governed by a strong interaction between Fe1 and Fe2. The lowest calculated value of $T_C^{\text{MFA}}$ = 1124.20 K at 30 GPa is mainly attributed to the low value of $J_{ij}$ due to the short range ($R_{ij}/a \sim 1$) interaction between Fe1 and Fe2 [see inset (blue line) in Fig. 8c]. For the Al-Fe1, Fe1-Fe1 and Co-Fe1 interactions, the $J_{ij}$ values are intermediate at 30 GPa [cf. Fig. 8a-d]. On increasing the pressure beyond 30 GPa there occurs a linear increase in $T_C^{\text{MFA}}$, which reaches a maximum at 60 GPa. Our results for $T_C^{\text{MFA}}$ contradict with the results of Rambabu et al. for Co₂CrX (X = Al, Ga, In) below 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 field 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

$$T_C^{\text{cal}} = 23 + 181 \times M_t \quad (4)$$

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 justifiable. Therefore, we have taken the $M_t$ values calculated with GGA+U and substituted in eqn (4) to obtain the $T_C^{\text{cal}}$. The estimated values of $T_C^{\text{cal}}$ are presented in Table 3. The $T_C^{\text{cal}}$ 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 $T_{CS}$ 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 $T_C^{\text{MFA}}$ may arise due to inability to include the magnetic percolation effect within the mean field approximation.⁵⁵

## 4 Conclusion

In summary, we have studied the structural, electronic and magnetic properties of the inverse HA Fe₂CoAl along with the $T_C$

<figure><img src="image_16.png" /><figcaption>Fig. 8 Exchange interaction $J_{ij}$ (meV) between (a) Al and Fe1, (b) Fe1 and Fe1, (c) Fe2 and Fe1 and (d) Co and Fe1 at different pressures.</figcaption></figure>
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 Fe₂-CoAl, ~4.0 µB at 30 GPa, is in accordance with the Slater-Pauling rule, which supports the half-metallicity. The Tc 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 conflicts to declare.

## Acknowledgements

D. P. R. acknowledges the Department of Science and Technology New Delhi, Govt of India *vide* Lett. No. INT/RUS/RFBR/P-264. I. V. M. acknowledges the Russian Foundation for Basic Research (RFBR), Russia, RFBR-17-57-45024. M. P. G. thanks the Alexander von Humboldt Foundation for the equipment subsidy grants. D. P. R. extends condolences to the family of Prof. G. M. Mikhailov, an Indo-Russian project collaborator.

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