First-principles investigation of half-metallic ferromagnetism of Fe$_2$YSn (Y = Mn, Ti and V) Heusler alloys

In this paper, we use the first-principles calculations based on the density functional theory to investigate structural, electronic and magnetic properties of Fe$_{2}$YSn with (Y = Mn, Ti and V). The generalized gradient approximation (GGA) method is used for calculations. The Cu$_{2}$MnAl type structure is energetically more stable than the Hg$_{2}$CuTi type structure. The negative formation energy is shown as the evidence of thermodynamic stability of the alloy. The calculated total spin moment is found as 3$\mu_\text{B}$ and 0$\mu_\text{B}$ at the equilibrium lattice constant for Fe$_{2}$MnSn and Fe$_{2}$TiSn respectively, which agrees with the Slater-Pauling rule of $M_t= Z_t-24$. The study of electronic and magnetic properties proves that Fe$_{2}$MnSn and Fe$_{2}$TiSn full-Heusler alloys are complete half-metallic ferromagnetic materials.


Introduction
In 1983, de Groot et al [1] established the half-metallicity in NiMnSb half Heusler alloy. Later, through theoretical calculations and experiments, many compounds were found to be half-metals, including Heusler alloys [2,3], alkali metal or transition metal chalcogenides [4], doped diluted magnetic semiconductors [5,6], zinc-blende and wurtzite structural compounds [7]. Considering the standing great potential advantages, spintronics still faces some challenges, such as generation of high spin injectors [8]. In recent years, Heusler alloys received an insistent attention due to their interesting physical properties [9][10][11], their remarkable electronic structure makes it possible to use them in various spintronic devices such as spin-transfer torque and large magneto-resistance spinvalves devices [12]. Heusler alloys are ternary inter-metallic compounds, which were first discovered by Heusler in 1903 [13]. This remarkable material and its relatives, which by now comprise a vast collection of more than 1000 compounds, are now known as Heusler compounds. They are ternary semiconducting or metallic materials with a 1:1:1 (also known as "half-Heusler") or a 2:1:1 stoichiometry (also known as "full-Heusler") [13]. Several Fe-based Heusler alloys have already been studied, though due to the differences in their experimental and theoretical results further investigations are still being carried out [3,13]. This paper is structured as follows: in section 2, we briefly describe the computational method used in this work. Results and discussions of our study are presented in section 3. Finally, a summary of the work is given in section 4.

Method of description
The first-principle calculations of Fe 2 YSn (Y = Mn, Ti, and V) alloys are performed based on the density functional theory (DFT) [14], which is implemented in WIEN2k code [15]. The solution of the Kohn-Sham equation [14] is done using the full potential linearized augmented plane wave (FP-LAPW) method [15]. The exchange correlation potential is calculated using the Perdew-Burke-Ernzerhof parameterization of the generalized gradient approximation PBE-GGA [16]. In the calculations reported in this paper, we use a parameter × max = 8, which defines the matrix size convergence, where max is the plane wave cut-off and is the smallest of all atomic sphere radii. In the full potential scheme, the whole crystal is divided into two different parts: the first part is the atomic sphere while the second part includes the interstitial regions. Moreover, the valence wave function inside the muffintin (MT) sphere was expanded up to max = 10, while the charge density was Fourier expanded up to max = 12 a.u −1 . The self-consistent calculations are considered to be converged when the total energy of the system is stable within 10 −4 Ry.

Results and discussions
There are three distinct families of Heusler compounds: the first one with the composition 1:1:1 and the second one with 2:1:1 stoichiometry and the third is 1:1:1:1. The compounds of the first family have the general formula XYZ and crystallize in a non-centro symmetric cubic structure; the second family of Heusler alloys has a formula X 2 YZ with two types of structures the Hg 2 CuTi and Cu 2 MnAl. The two phases consist of four inter-penetrating fcc sub-lattices, which have four crystal sites, 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 Hg 2 CuTi type structure, the chain of atoms occupies the four sites of unit cell X-X-Y-Z and for Cu 2 MnAl the Y and the second X atom exchange sites. In the Hg 2 CuTi type structure, the X atoms entering sites A and B are denoted as X(1) and X(2), respectively [13]. The third family has a formula of XX'YZ and crystallize in the LiMgPdSn type crystal structure. For the Heusler alloys X 2 YZ, the X and Y are both a transition metal, and Z is the main group element. In order to establish a stable structure and equilibrium structural parameters of Fe 2 ZSn (Z = Mn, Ti and V) compounds, structural optimizations were performed on these alloys for both Cu 2 MnAl and Hg 2 CuTi type structures and their total energy-volume curves are shown in figure 1. In X 2 YZ Heusler alloys, if the Y atomic number is superior to that of X atom from the same period, an inverse Heusler structure with Hg 2 CuTi type as the prototype is observed. It is seen from these E-V curves that the Cu 2 MnAl type structure is more stable than the Hg 2 CuTi phase for the Fe 2 YSn with Y = Mn, Ti, V compounds at ambient conditions. The nuclear charge of X atom (Fe) is larger than Y atom (Y = Mn, Ti and V). Consequently, the Cu 2 MnAl structure will be visibly observed as can be seen from figure 1. The minimum of the curve is the calculated equilibrium lattice constant. The lattice constant , bulk modulus and its pressure derivative at zero pressure, for the structures Cu 2 MnAl and Hg 2 CuTi are calculated using Murnaghan equation of state [17].
Here 0 is the minimum energy at = 0 K, is the bulk modulus, is the bulk modulus derivative and 0 is the equilibrium volume. The results are listed in table 1. The calculated lattice constants of Fe 2 YSn with (Y = Mn, Ti and V) are in good agreement with the previously theoretically optimized lattice constants reported by other researchers.
We study the phase stability of Fe 2 YSn with (Y = Mn, Ti, V) based on the formation energy (Δ ). This can help to envisage whether these alloys can be prepared experimentally. Here, the formation energy (Δ ) is calculated by comparing the total energies of the Fe 2 YSn (Y = Mn, Ti, and V) Heusler alloys with the sum of the total energies of the constituting elements. The formation energy of the Fe 2 YSn (Y = Mn, Ti, and V) materials is computed following the expression given below:    [24]. The -band is principally responsible for the position of Fermi level lying in it. The responsibility of transition metals 3 -states is very important in the description of spin polarized electronic band structures and densities of sates calculations [24]. At the equilibrium lattice constants, we have studied the electronic band structure calculations for all three compounds and have extracted the density of states (DOS) per f.u., which is presented in figure 2, figure 3 and figure 4. The electronic band structure shows the bonding and character of the electron bands. DFT is a standard tool for calculating the band structure for materials in order to determine different properties of solids [25,26]. The responsibility of transition metals 3 -states is very essential in the description of spin polarized electronic band structures and densities of sates calculations [24]. The electron spin polarization (SP) at EF of a material is defined as follows [23] = ↑ ( ) − ↓ ( ) ↑ ( ) + ↓ ( ) .

(3.3)
Here ↑ ( ), ↓ ( ) are the majority and minority densities of states at . When the value of the electron spin polarization (SP) is 100%, alloys are supposed to be true half-metallic, and this is realized when any one of the DOS from the majority and minority spins is equal to zero and the other one is not equal to zero at EF [23]. Figure 2 presents the total density of states and band structure of Fe 2 MnSn for both Cu 2 MnAl and Hg 2 CuTi type structures. From this figure, one can observe that the alloy exhibits half-metallic behavior with Cu 2 MnAl type structure, described by an overlap between the bottom of the conduction band and the top of the valence band in spin-up. For spin-down, we can see a gap between the maximum of the valance band and the minimum of the conduction band, this gap being indirect between Γ and X points. For Hg 2 CuTi type structure, both the majority and minority spin bands have metallic intersections at the Fermi level. We can see from figure 3 that the total DOS in spin-up and spin-down spin channels for Fe 2 TiSn, is symmetrical in the majority and minority spin directions. Therefore, the non-magnetic character of these alloys can be estimated. In both spin directions, the energy gap is open

23703-4
First-principles investigation of half-metallic ferromagnetism  [27]. At the same time, the Sn 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 − hybridization between the valence states of Fe and Y atoms (Y = Mn, Ti and V). Skaftouros et al. [28] have presented fascinating arguments regarding possible hybridizations between -orbitals of transition metals in the case of the X 2 YZ Inverse Heusler compounds, e.g., Sc-based Heusler compounds. According to their report, the same symmetry of the X [1] and the Y atoms causes their -orbitals to hybridize together creating five bonding (2 × and 3 × 2 ) and five non-bonding (2 × and 3 × ) states. Then, the five X(1)-Y bonding states hybridize with the -orbital of the X(2) atoms and create bonding and anti-bonding states again (3).   We present the total magnetic moment, the local magnetic moments on Fe, Y (Y = Mn, Ti, and V), Sn atoms and interstitial moments which are given per unit cell. The calculated local and total magnetic moments in interstitial region for a Heusler compound Fe 2 YSn with (Y = Mn, Ti and V) are presented in table 2. It must be noted that the total magnetic moment is very sensitive to both types of structures. For Fe 2 YSn with (Y = Mn, Ti and V) with Hg 2 CuTi type structure, the magnetic moment is mostly located

23703-6
First-principles investigation of half-metallic ferromagnetism  on the Fe atoms. Indeed, a great part of the total magnetic moment results from this atom. Therefore, the Sn atom has a minor magnetic moment, which does not give a lot to the total magnetic moment.
The found values of the magnetic moment are dependent on the Slater-Pauling curve (SPC) [30] for full Heusler alloys, in which the magnetic moment per unit cell in multiples of Bohr magnetons ( B ) can be calculated as follows: Here tot represents the total magnetic moment and N represents the total valence electrons in the unit cell. is equal to 27 for Fe 2 MnSn [(8 × 2) + 7 + 4 = 27], it is equal to 24 for Fe 2 TiSn and [(8 × 2) + 4 + 4 = 24]. The value of calculated magnetic moment have an integer value of 0 B and 3 B for Fe 2 TiSn and Fe 2 MnSn respectively with Cu 2 MnAl type structure, which matches well with moments predicted from Slater-Pauling rule and evidences that these have potential to be half metallic. The magnetic moment of Fe 2 TiSn is zero, the interaction between 3 electron of Fe and 3 electrons of Ti are in opposite directions and cancel the over-all moment. The Heusler alloy Fe 2 TiSn has 24 valence electrons. These electrons occupy the majority and minority spin bands equally (12 up and 12 down), which results in a nonmagnetic semiconductor-like band structure [29]. These results agree well with preceding studies in Fe 2 TiSn [29]. The total magnetic moment of Fe 2 VSn is equal to 2.75866 B and 3.23143 B for Hg 2 CuTi and Cu 2 MnAl type structure very far from the integer value witch confirm that this alloys have not half metallic behavior.

Conclusion
We have performed ab-initio calculations to investigate the structural, electronic, magnetic properties of Fe 2 YSn (Y = Mn, Ti and V) Heusler alloys with both Cu 2 MnAl and Hg 2 CuTi type structure. The negative formation energy is shown, as an evidence of the thermodynamic stability of Fe 2 YSn (Y = Mn, Ti, and V) alloy. The Cu 2 MnAl type structure is energetically more stable than the Hg 2 CuTi type structure. Our calculations indicate that the 6.08 Å, 5.98 Å and 5.95 Å are the equilibrium lattice constant of Fe 2 TiSn, Fe 2 VSn and Fe 2 MnSn with Cu 2 MnAl, respectively. Furthermore, Fe 2 MnSn and Fe 2 TiSn in the ground state is considered to be a true half-metallic based on the calculations of the band structure and density of states. It is also predicted that Fe 2 MnSn and Fe 2 TiSn compounds are half-metallic with 100% spin polarization with an integer magnetic moments making these compounds a good candidates for spintronic devices applications.

Acknowledgement
This work has been supported by DGRST-ALGERIA.