# The effect of Sn substitution on the Al sites in full Heusler compound Fe_{2}VAl

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## Abstract

A first-principles method has been employed to determine the electronic, magnetic and structural characteristics of the full-Heusler alloys Fe_{2}VAl with Sn doping. All the calculations were performed by using a computational code based on full-potential linearized augmented plane wave method called WIEN2k. The electron exchange–correlation is treated by the generalized gradient approximation within a scheme developed by Perdew, Burke and Ernzerhof (PBE-GGA). The electronic band structures of Fe_{2}VAl_{1−x}Sn_{x} (x = 0, 0.25, 0.50, 0.75) compounds show that the majority-spin (spin-up) exhibits a metallic characteristic, whereas the minority-spin (spin-dn) have an energy band gap. Our calculations predict that Fe_{2}VAl_{1−x}Sn_{x} compounds are half-metallic ferromagnets with an integer value of magnetic moment, 0, 1, 2, and 3 μ_{B}, respectively. Our findings suggest that these materials are potential candidates for manufacturing spintronic devices.

## Keywords

Heusler alloys Half-metallicity Magnetic moment First principle calculations## PACS Nos.

71.15.Mb 71.20.−b 75.90. +w## 1 Introduction

Full-Heusler alloys have a general chemical formula *X*_{2}*YZ*, where *X* and *Y* are d-block elements and *Z* is a main group atom; in several cases, Y is a rare earth or an alkaline earth metal. The full-Heusler alloys takes the stoichiometric ration 2:1:1 in which the heavier atom (high atomic number) occupies the two X sites while the other two nonmagnetic atoms occupy Y and Z sites in a primate cell such as Fe_{2}VAl [1]. Full-Heusler crystallizes in cubic *L*_{21} phase. The formation of magnetic moment and magnetic coupling in X_{2}YZ-type full-Heusler compounds has been studied by Kübler et al. [2]. Heusler compounds are considered to be important materials due to their wide range of phenomenal multifunctional properties such as half-metallic ferromagnetic (HMF) and high Curie temperature, which make them suitable candidate for electron spin-based futuristic technological devices [3, 4, 5, 6]. The existence of band gap around the Fermi level (*E*_{F}) for the spin-down states whereas the spin-up states are occupied with free valence electron that gives the dispersed DOS at *E*_{F}. Hence, HMF is a peculiar metal–semiconductor hybrid which indicates 100% spin polarization at the Fermi energy. The half-metallic materials are often considered as a potential candidate in spintronic applications [7]. NiMnSb is a prototype half-metal half-Heusler alloy discovered by de Groot et al. [8] in 1983. In the last decade, the Fe_{2} based full Heusler has been the topic of interest due to its wide variety of extraordinary transport and magnetic properties such as semimetals, semiconductors, half-metallic ferromagnetic behavior, weakly ferromagnetic and antiferromagnetic materials [9, 10, 11, 12, 13, 14, 15, 16, 17].

In this work, we have investigated the structural parameters, electronic structure and half-metallic ferromagnetic property of Fe_{2}VAl_{1−x}Sn_{x} (x = 0, 0.25, 0.50, 0.75 and 1) Heusler alloys by employing the density functional theory (DFT) calculations. The paper is organized as follows: In Sect. 2, we briefly describe the computational techniques used in this study. The most relevant results obtained for the structural, electronic and magnetic properties of Fe_{2}VAl_{1−x}Sn_{x} (x = 0, 0.25, 0.50, 0.75 and 1) Heusler compounds are presented and discussed in Sect. 3. Finally, in Sect. 4 we summarize the main conclusions of our work.

## 2 Calculation methods

In the current study, the calculations were carried out by using the full-potential linearized augmented plane wave (FP-LAPW) method based on the density functional theory (DFT) [18, 19] as implemented in the Wien2k code [20]. The exchange–correlation potential was treated by the generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhof [21]. In FP-LAPW method the space is divided into an interstitial region (IR) and non overlapping muffin tin (*MT*) spheres centered at the atomic sites. In the IR region, the basis set consists of plane waves. Inside the *MT* spheres, the basis sets is described by radial solutions of the one particle Schrödinger equation (at fixed energy) and their energy derivatives multiplied by spherical harmonics. The wave functions in the interstitial region were expanded in plane waves with a cut-off *K*_{max} = 8/*R*_{MT}, where R_{MT} denotes the smallest atomic sphere radius and *K*_{max} is the maximum modulus for the reciprocal lattice vectors. The *R*_{MT} are taken to be 2.21 atomic units (a.u.) for Fe and V and 2.07 (a.u.) for Al in Fe_{2}VAl, and 2.30 (a.u.) for Fe and V and 2.16 (a.u.) for Sn in Fe_{2}VSn. The valence wave functions inside the spheres are expanded up to *l*_{max} = 10, while the charge density was Fourier expanded up to *G*_{max} = 14. The cut-off energy which separates the valence and core states was selected as − 7.5 Ry. A convergence norm for self-consistent field calculations was chosen in such a way that the difference in energy between two successive iterations did not exceed 10^{−4} Ry. The electronic configurations for the valence states of Fe, V, Sn and Al are Fe: 3d^{6}4s^{2}, V: 3s^{2}3p^{6}3d^{3}4s^{2}, Sn: 5s^{2}5p^{2} and Al: 3s^{2}3p^{1}.

*E*

_{0}is the minimum energy at the equilibrium volume

*V*

_{0},

*B*is the bulk modulus and B’ is pressure derivative of the bulk modulus. To simulate the quaternary alloys Fe

_{2}VAl

_{1−x}Sn

_{x}(x = 0.25, 0.50, 0.75), we have generated a supercell which generates 8 Fe atoms, 4 V atoms and 4 Al atoms, from the most stable structure. For x = 0.25 we replaced one atom of Al by one atom of Sn, for x = 0.50, we replace 2 atoms of Al by 2 atoms of Sn, and for x = 0.75, we replace 3 atoms of Al by 3 atoms of Sn.

## 3 Results and discussions

*L*

_{21}structure, with two phases Cu

_{2}MnAl and Hg

_{2}CuTi structures. The two structures consist of four inter-penetrating fcc sub-lattices, which have four unique crystal sites, i.e., A (0, 0, 0), B (0.25, 0.25, 0.25), C (0.50, 0.50, 0.50), and D (0.75, 0.75, 0.75). For Cu

_{2}MnAl structure the chain of atoms occupy the four sites of the unit cell is

*X*–

*Y*–

*X*–

*Z*, and for the Hg

_{2}CuTi structure, the Y and the second X atom exchange sites and the chain of the atoms is

*X*–

*X*–

*Y*–

*Z*. In the Hg

_{2}CuTi-type structure, the X atoms entering sites A and B are denoted as X (1) and X (2), respectively. The most important difference between these two structures is the exchange between the C site atom and the B site atom. Both structures may be indistinguishable by X-ray diffraction and both have the general fcc-like symmetry [23].In X

_{2}YZ compounds, if the Y atomic number is higher than that of the X atom from the same period then it is called an inverse Heusler structure (F43m, space group no. 216) with Hg

_{2}CuTi-type as the prototype is observed [24]. To confirm this site preference law and to determine the structural properties at the stable structure of Fe

_{2}VAl and Fe

_{2}VSn compounds, we have calculated total energies as a function of volumes of these alloys in both Cu

_{2}MnAl- and Hg

_{2}CuTi-type structures. The computed total energies versus volumes are plotted in Fig. 1. It is seen that Cu

_{2}MnAl-type has minor energy, indicating that this structure is stable compared to the Hg

_{2}CuTi-type structure, and in Fe

_{2}VAl and Fe

_{2}VSn, the Y (V) atomic number is 23, inferior to the X (Fe) atomic number of 26; for this reason, the Cu

_{2}MnAl type as a prototype is observed in Fig. 1, and the Fe atoms prefer to occupy the A and C sites instead of the A and B sites.

_{2}VAl

_{1−x}Sn

_{x}(x = 0, 0.25, 0.50, 0.75 and 1) is obtained by fitting the Murnaghan equation of state [22] and presented in Fig. 1. The total energy versus volume curve determines the optimized parameters like lattice constant the bulk modulus and its pressure derivatives. The computed results are listed in Table 1. The obtained lattice parameters are found to be 5.82 and 6.06 Å for the Fe

_{2}VAl and Fe

_{2}VSn, respectively with Hg

_{2}CuTi type structure and 5.71 and 5.99 Å with Cu

_{2}MnAl type structure, which are in excellent conformity with previous theoretical calculations [25].The calculated lattice parameter increases with the addition of Sn to alloy, consequently volume increases and we can see also from Table 1 that bulk modulus decreases with the addition of Sn.

Lattice constant a (Å), bulk modulus B (GPa) for Fe_{2}VAl_{1−x}Sn_{x} (x = 0, 0.25, 0.50, 0.75, 1)

Compound | Type structure | a (Å) | B (GPa) | Volume (a.u.) | Energy (Ry) |
---|---|---|---|---|---|

Fe | Hg | 5.82 | 238.2689 | 332.0605 | − 7475.530858 |

Cu | 5.71 | 222.9529 | 313.8353 | − 7475.618656 | |

5.7129 [12] | 216.3 [12] | ||||

5.7166 [25] | 230.54 [25] | ||||

Fe | Hg | 6.06 | 374.2512 | 373.1921 | − 19,348.051962 |

Cu | 5.99 | 302.4447 | 363.3512 | − 19,348.079892 | |

5.95 [3] | |||||

5.9688 [7] | |||||

Fe | 5.79 | 197.6658 | 1302.5383 | − 41,774.906317 | |

Fe | 5.85 | 261.9867 | 1355.4023 | − 53,647.362008 | |

Fe | 5.93 | 264.5033 | 1406.6815 | − 65,519.846428 |

_{2}VAl

_{1−x}Sn

_{x}(x = 0, 0.25, 0.75), and hence we have compared the lattice constants to the Vegard’s law given by Eq. (2).

*(↑*or

*↓)*contributes to the conductivity. This is the case explained by de Groot et al. [8]: the densities of states have a gap at Fermi level for one spin direction, and in Type Ia half-metals, the half-metallic gap appears in the minority density of states. The electrons at E

_{F}are itinerant in Type I half-metals, while they are localized in Type II. In Type III half-metals, no gap is shown at the Fermi energy, but one has at

*E*

_{F}itinerant electrons for one spin direction and localized electrons for the other [27, 28]. The spin polarization at

*E*

_{F}is given by:

*N↑*(

*E*

_{F}) and

*N↓*(

*E*

_{F}) are the spin-dependent densities of states at E

_{F}. The arrows ↑ and ↓ assign states of the opposite spins, corresponding to the majority and minority states, respectively.

_{2}VAl and Fe

_{2}VSn alloys with both Hg

_{2}CuTi and Cu

_{2}MnAl structures are plotted in Figs. 2 and 3 with GGA approximation and GGA + U approximation, respectively. In Fig. 2, the spin up and spin down bands are represented by blue and line, respectively. Similar representations are used to denote the spin up and spin down states in the total density of states (DOS).For Fe

_{2}VAl with the Cu

_{2}MnAl structure (Fig. 2), both spin up and spin down bands show a miniature band gap at the gamma (Γ) point indicate a semiconducting behavior. At the X point, conduction and valence bands both overlap, signifies metallic behavior. We can see that the DOS of Fe

_{2}VAl with the Cu

_{2}MnAl-type structure (Fig. 3), both the spin up and spin down states show a pseudo gap at E

_{F}, to the approximately zero density of electron charge. This is a typical semi-metallic characteristic, where the spin polarization for Fe

_{2}VAl is negligible because of the DOS at E

_{F}in both the spin states is missing [21]. The pseudo-gap of Fe

_{2}VAl is formed by the strong correlation between Fe-3d and V-3d levels. In Fe

_{2}VAl with the Hg

_{2}CuTi-type structure (Figs. 2 and 3), we can see the metallic character of this compound. The electronic band structures and the DOS of the Fe

_{2}VSn compounds for both the Hg

_{2}CuTi and Cu

_{2}MnAl type structures are given in Figs. 2 and 3, and the DOS of both spin up and spin down occur at Fermi level; as a result, Fe

_{2}VSn has a metallic character. Both the conduction and valence band crosses the Fermi level, thus diminishing the gap at E

_{F}. The metallic nature in this compound is principally due to the interaction between Sn-p and transition metal (TM)-3d states. In the spin up channel, the presence of DOS at E

_{F}is significantly small. Mahmoud et al. [7] reported that the ideal Fe

_{2}VSn shows a vanishing band gap, owing to the overlap between the conduction and valance bands that gives 80% spin polarization at E

_{F}. The band gap (in majority or minority spin), being such a symbolic character for the half-metallic Heusler alloy, must be taken into account to clarify the source of the band gap. It is known that the origin of the band gap is always divided into three categories: (1) covalent band gap, (2) d–d band gap, and (3) charge transfer band gap [29]. The covalent band gap has been found to exist in half- Heusler alloys with a C1

_{b}structure, such as NiMnSb [30]. The d–d band gap is responsible for the half-metallic character of the full-Heusler alloys with an L

_{21}structure. Generally, the origin of the band gap in a Heusler alloy is a result of bonding antibonding states, ascribed to the hybridization between the lower-energy d-orbitals of the high-valence transition metal and the higher-energy d-orbitals of the low-valence transition metal [31]. In Fig. 4, the energy gap of Fe

_{2}VAl

_{1−x}Sn

_{x}(x = 0.25, 0.50, 0.75) exists in the spin down, while the bands of majority spin channel occur at E

_{F}. In other words, the spin up states show a metallic nature, whereas the minority spin bands have a gap but not at the Fermi level (E

_{F}), which is a semiconducting feature; this suggests that Fe

_{2}VAl

_{1−x}Sn

_{x}(x = 0.25, 0.50, 0.75) compounds are half metals (HM) of Type III with energy gaps of 1.2 eV, 0.82 eV and 0.44 eV for x = 0.25, 0.50, and 0.75, respectively.

_{2}VAl

_{1−x}Sn

_{x}are summarized in Table 2; these values were computed based on the Al and Sn concentrations.

Calculated total and local magnetic moment (in μ_{B}) of Fe_{2}VAl_{1−x}Sn_{x} (x = 0, 0.25, 0.50, 0.75 and 1)

Compound | Type structure | Approximation | M | M | M | M | M | M | M |
---|---|---|---|---|---|---|---|---|---|

Fe | Hg | GGA | 1.96011 | 1.92995 | − 0.19277 | − 0.03198 | – | − 0.07599 | 3.58933 |

GGA + U | 1.38228 | 2.5980 | − 0.73732 | − 0.05007 | − 0.16313 | 3.02982 | |||

Cu | GGA | 0.00 | 0.00 | 0.00 | 0.00 | – | 0.00 | 0.00 | |

GGA + U | 0.00 | 0.00 | 0.00 | 0.00 | – | 0.00 | 0.00 | ||

0.00 [20] | 0.00 [20] | 0.00 [20] | 0.00 [20] | 0.00 [20] | |||||

Fe | Hg | GGA | 1.59871 | 2.54,979 | − 1.30225 | − 0.00688 | − 0.24919 | 2.59018 | |

GGA + U | 1.21325 | 2.97699 | − 1.39522 | − 0.01373 | − 0.27773 | 2.50356 | |||

Cu | GGA | 0.78831 | 0.78823 | − 0.43661 | – | − 0.01508 | − 0.14902 | 0.97582 | |

− 0.50747 [32] | – | 0.02098 [32] | 0.01373 [32] | 0.97681 [32] | |||||

3.64 (unit cell 16 atoms) [7] | |||||||||

Fe | 0.0739 | 0.0741 | 0.1125 | 0.1065 | 0.13259 | − 0.02225 | 0.99232/(unit cell 16 atoms) | ||

Fe | 0.2883 | 0.2884 | − 0.0148 | − 0.0183 | − 0.01020 | − 0.20730 | 1.98276 (unit cell 16 atoms) | ||

Fe | 0.4741 | 0.4939 | − 0.1275 | − 0.02470 | − 0.01338 | − 0.31307 | 2.98405 (unit cell 16 atoms) |

Slater and Pauling showed that for the binary magnetic alloys, while one valence electron is added to the alloy, it resides in spin-down states, and the total spin magnetic moment is reduces by 1.0 μ_{B} [33, 34], analogues behavior can also be establish in half-metallic Heusler alloys. For the semi-Heusler compounds, for example NiMnSb, the total spin magnetic in the unit cell Mt, change as a function of the total number of valence electrons *Zt*, following the relation *Mt *=* Zt *−* 18* [30], whereas in the case of the L_{21} full Heusler, this relation becomes *Mt *=* Zt *−* 24* [31]. Slater–Pauling (SP) rules join the electronic properties (the half-metallic behavior) directly to the magnetic properties (total magnetic moments) and thus present a great tool for the study of half-metallic Heusler alloys. Additionally to the magnetic moment localized at V atom at the cubic site, the Fe atoms at the tetrahedral sites interact with one another, which may result in a supplementary magnetic interaction between the Fe atoms. In the case of four atoms per formula unit, as is the case in Fe_{2}VAl and Fe_{2}VSn compounds, 24 should be subtracted from the number of valence electrons per formula unit to find the magnetic moment per formula unit.

The Fe_{2}VAl alloy has 24 valence electrons per unit cell (*Z*_{t} = (8 × 2) + 5+3 = 24), so the spin magnetic moment is 24 − 24 = 0. We have noticed half-metallic character is also confirmed by the integer value of total magnetic moment [35, 36]. Many studies found that half-metallic Heusler alloys have integral total magnetic moments, which follows the Slater–Pauling rule [31, 36]. From Table 2, the computed magnetic moments are in good agreement with other theoretical calculations. The magnetic moment is approximately linearly proportional with the concentration of Sn atoms as well as the number of valance electrons in the unit cell; this magnetic moment tends to be augmented with increasing Sn concentration: (0, 1, 2, and 3 μ_{B}) for x = 0, 0.25, 0.50, and 0.75, respectively.

## 4 Conclusions

We have calculated the structural parameters, electronic and magnetic properties of ternary (Fe_{2}VAl, Fe_{2}VSn) and quaternary (Fe_{2}VAl_{1−x}Sn_{x} with x = 0, 0.25, 0.50, and 0.75) Heusler alloys by using first-principle method of density functional theory. The quaternary Heusler alloys Fe_{2}VAl_{1−x}Sn_{x} revealed a true half-metallic behavior with a gap in minority-spin states and spin polarization of 100%. Therefore, Fe_{2}VAl_{1−x}Sn_{x} (x = 0, 0.25, 0.50, and 0.75) compounds seem to be good candidates to explore half-metallic ferromagnetic feature in practical applications for spintronic devices.

## Notes

### Acknowledgements

For the authors “R.K and S.B.O” this project was funded by the National Plan for Science, Technology and Innovation (MAARIFAH) King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, Award Number: 11-NAN1465-02.

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