Ab initio study of magnetic and structural properties of Fe-Ga alloys

The structural and magnetic properties for a series of Fe100-xGax alloys (x = 18 – 30 at.%) are studied in the framework of first-principles calculations and Monte Carlo simulations. The both, general gradient approximation and local density approximation are considered for the exchange-correlation functional. The ground state ab initio calculations are performed for both D03 and L12 crystal structures. It is shown that for general gradient approximation, the optimized lattice parameters and total magnetic moments are found in the better agreement with experimental ones. Using the calculated exchange coupling constants for studied compositions, Curie temperatures are estimated by means of Monte Carlo simulations of Heisenberg Hamiltonian.

The aim of this paper is a complex study of magnetic and structural properties of Fe 100-x Ga x (x = 18 -30 at%) alloys by the density of functional theory with the different approximations for the exchange-correlation energy and finite-temperature Monte Carlo (MC) simulations.

Computation details
In the first step of our calculations, we have done the geometric optimization of crystal structures by using the SPR-KKR (spin polarized relativistic Korringa-Kohn-Rostoker) software package based on the Korringa-Kohn-Rostoker Green's function [17]. For all ab initio calculations, the exchange-correlation energy was treated by the generalized gradient approximation in the Perdew-Burke-Ernzerhof (GGA-PBE) formulation [18] and the local density approximation in the form of Vosko-Wilk-Nusair (LDA-VWN) [19]. For selfconsistent cycles (SCF) calculations, 6348 k points were generated by a k-mesh grid of 45 3 . The total energy for all calculations converged to 0.01 mRy. To construct offstoichiometric compositions, the CPA was considered.
To perform the crystal structure optimization, we used two face-centered cubic L1 2 (Pm3 � m, #221) and body-centered cubic D0 3 (Fm3 � m, #225) structures with their associated unit cells, which contain 4 atoms. Note that the stoichiometric Fe 75 Ga 25 (Fe 3 Ga) can crystallize in the D0 3 structure [6]. There are three Fe atoms per unit cell belonging to two different sublattices. For the formation of non-stoichiometric compositions of Fe 100- x Ga x , Ga atoms taken in the required percentage were fixed at next sites as listed in Table 1.
In the second step, using the optimized lattice parameters obtained within GGA-PBE and LDA-WVN approximations, the calculations of exchange interaction parameters J ij for both structures were performed. The Heisenberg exchange coupling constants were calculated using the expression proposed by Liechtenstein et al. [20]. All calculations converged to 0.01 mRy of total energy. Here The exchange parameters J ij and partial magnetic moments μ were taken as input parameters from ab initio calculations. The model lattice with periodic boundary conditions for Fe 75 Ga 25 alloy contains 2826 (3300) Fe and 1099 (1331) Ga atoms for D0 3 (L1 2 ) phase, respectively. The MC simulations were carried out using the Metropolis algorithm [21]. As time unit, we used on MC step consisting of N attempts to change the spin variables. The total number of MC steps per the temperature step was 10 6 . The magnetic order parameter m and total magnetization M are defined by the following way here μ Fe is the magnetic moment of Fe taken from first principals calculations.
To estimate the Curie temperature from M(T) curves, we plotted the M 1/β (T) function that decreases almost linearly with increasing temperature. The Curie temperature can be evaluated at the intersection of a M 1/β curve with the T axis. Here, β is the critical index and it is equal to 0.3646 for the three-dimensional Heisenberg model.

Lattice parameters and magnetic moments
In this subsection, we present the results of calculations of the optimized lattice parameters and total magnetic moments for studied compositions with D0 3 and L1 2 structures. Fig. 1 shows the variation of equilibrium lattice parameter a 0 (which corresponds to a minimum value of energy E 0 ) as a function of Ga concentration for both D0 3 and L1 2 structures of Fe 100-x Ga x alloys in comparison with available experimental data taken from Ref. [22]. Here we present results obtained by using GGA-PBE and LDA-WVN approximations for the exchangecorrelation functional. Our calculation have shown that for both D0 3 and L1 2 structures, lattice constant increases with increasing Ga concentration. For both bcc and fcc structures, the results obtained within the GGA-PBE approximation are in the better agreement with the experimental data than those obtained for the LDA-VWN approximation.
In Fig. 2, we present the calculated total magnetic moments for the studied compositions Fe 100-x Ga x (x = 18 -30 at%) with D0 3 and L1 2 structures. It is clearly seen that for both structures, the total magnetic moment decreases with increasing Ga content. Concerning the effect of choice for the exchange-correlation potential, we can see the similar trend as in Fig. 1. Namely, for both D0 3 and L1 2 structures, the values of magnetization obtained within the GGA-PBE approximation are closer to the experimental data taken from [8].
Generally, according to Ref. [23], exchangecorrelation functional GGA reproduces the equilibrium volume of 3d-metals better than the LDA. Believe that the LDA fails to correctly reproduce the ground state bcc structure of Fe because the magnetic contribution to the stabilization of the bcc structure is weakened by this exchange-correlation functional while at the same time the GGA corrects the volume and thereby the magnetic contribution. We could equally well say that for the ground state calculations of Fe-Ga alloys with different phases the method of Green's function within GGA-PBE gives better results than LDA-VWN.

Magnetic exchange interaction and Curie temperature
The knowledge of optimized lattice parameters allows us to calculate further the Heisenberg exchange interaction parameters J ij for both bcc and fcc structures of Fe 100- x Ga x alloys. Since the results of lattice parameter calculations obtained within the LDA-VWN differ significantly from the experimental ones, we further used only the set of a 0 calculated within the GGA-PBE. Nevertheless, the calculations of J ij parameters were calculated using both approaches: GGA-PBE and LDA-WVN. As a result, we will use two new denotations: GGA-GGA and GGA-LDA. The first (second) one denotes the J ij calculations within the GGA-PBE (LDA-WVN) using the set of a 0 calculated within the GGA-PBE, respectively. Fig. 3 displays the magnetic exchange parameters for D0 3 and L1 2 phases of the Fe 73 Ga 27 alloy as a function of the distance between atoms. The Fe 73 Ga 27 composition is of interest due to the presence of structural phase transition between D0 3 and L1 2 phases [4]. For both phases, the oscillating damped behavior of J ij can be observed. We can see that for D0 3 structure, the strongest ferromagnetic interaction (J ij ˃ 0) is found between nearest-neighbors Fe 1 -Fe 2 (which are located at 4b and 8c Wyckoff positions, respectively). In a case of L1 2 structure, the ferromagnetic contribution to the total exchange energy between nearest neighbors is found to be slightly smaller. Moreover, it is seen that in the second coordination shell (d/a = 1) for L1 2 structure, the Fe-Fe exchange parameters split into two FM contributions. In this case, each Fe at the distance of a 0 interacts ferromagnetically with six nearest neighbors: two Fe atoms provide J ij = 7.03 (7.14) meV (GGA (LDA)) and four Fe atoms locating in (x, y) plane provide J ij =2.59 (2.65) meV (GGA(LDA)), respectively. As can be seen, the difference between values J ij obtained with two approximations is rather small, especially for L1 2 phase (see the inset in Fig. 3b). The knowledge of the constants of magnetic exchange interactions and magnetic moments allows us to simulate the temperature dependences of magnetization and to estimate the Curie temperatures for both investigation structures by means of Monte Carlo simulations of the classical Heisenberg Hamiltonian without a magnetic anisotropy term. Fig. 4 shows the results of evaluations of the Curie temperature for the crystal structures of D0 3 and L1 2 of Fe 100-x Ga x alloys. In general, theoretical results of Curie temperature estimations are in a good agreement with experimental data [4,22]. It should be noted that the using of the magnetic exchange interactions obtained for GGA-LDA approach gives the results, which are the closer to the experimental data. Fig. 4. The Curie temperature for the crystal structures (a) D0 3 and (b) L1 2 of Fe 100-x Ga x alloys in comparison with experimental data, which were taken from [4,22].

Conclusion
In this work, we have introduced ab initio calculations and Monte Carlo simulations to study the structural and magnetic properties of D0 3 and L1 2 phases of Fe 100-x Ga x (x = 18 -30%). The geometric optimization of D0 3 and L1 2 structures and calculations of exchange magnetic interactions have been performed by using the SPR-KKR package with treated of exchange-correlation energy in different approximations. It is shown that for the ground state calculations of both D0 3 and L1 2 structures of Fe-Ga alloys, the using of the GGA-PBE functional gives better results than LDA-VWN. However, The LDA-VWN approximation is favored for calculations of the exchange parameters J ij and estimation of Curie temperatures by using the MC simulations of Heisenberg model. For the latter case, the theoretical values of Curie temperature of studied compositions for both phases are found in the better agreement with experimental ones.