Ab initio investigation of impurity ferromagnetism in the Pd1-xFex alloys: concentration and position dependence

We present the ab initio results of the structural and magnetic properties of the Pd host matrix implanted with Fe solute atoms at various concentrations. By means of density functional theory we confirm that iron impurities are able to initialize significant magnetization of the Pd atoms, when the impurity consentation exceeds 3 at.%. Besides, we demonstrate that the imposed magnetization depends on impurity positions in the host matrix, in particular, there is a maximum of magnetization for a uniform distribution of the iron impurity.

There is a scatter of opinions on compositional and magnetic homogeneity of Pd 1−x Fe x films obtained by different deposition techniques: some papers report on an indication of clustering [10,11,12,14,15] or formation of the nanograins of Pd 3 Fe phase [16,17] in samples deposited with the magnetron sputtering technique; the others on MBE growth of thin Pd 1−x Fe x films [20,21,22,23,24,25] report on their high magnetic homogeneity; finally, the ion-implanted palladium films [26] show a significant influence of the Fe implant distribution on saturation magnetization (M s ) and Curie temperature (T C ) of the samples. It is not clear why and how the redistribution of iron solute in the Pd matrix influences M s and T C . From very few existing studies of the electronic structure of Pd 1−x Fe x solid solutions, the Korrringa-Kohn-Rostoker Green's function method with the local-density functional approximation was used to caltulate the magnetic moment at the palladium site in the dilute substitutions (x close to 1.0) of Pd in bcc Fe [27], while the electronic structures and magnetic properties of Pd 1−x Fe x alloys with 0.5 ≤ x ≤ 0.85 were investigated in the framework of density functional theory (DFT) using the full potential approximation [28]. The only in a Pd-rich side is Ref. [29], where the magnetic properties of palladium-iron alloys and compounds were calculated by means of spin-polarized and scalar relativistic tight-binding linear muffin tin orbital method (TB-LMTO) within atomic sphere approximation (ASA), together with the coherent potential approximation (CPA) to describe random Pd-Fe solutions. However, magnetic moments and Curie temperatures at x < 0.1 were not presented, therefore, the calculations of Ref. [29] are not applicable to Pd-rich alloys.
In this work, we perform ab initio calculations for the Pd 1−x Fe x alloy with the iron dopant uniformly distributed over the bulk Pd host lattice at the different iron content x < 0.1 and calculate the mean magnetic moment per Fe solute atom and the maximal magnetic moment located on a Pd atom as a function of x. Then, we study the influence of inhomogeneity in the iron dopant distribution on the magnetic moment per Fe atom and the size and shape of the magnetized "bubble" of the host Pd atoms around the Fe aggregates.

Computational details
Our ab initio investigations were based on the DFT [30,31] approach within the VASP code [32,33,34] as a part of the MedeA ® software of Materials Design [35]. Exchange and correlation effects were accounted for by the generalized gradient approximation (GGA) as parameterized by Perdew, Burke, and Ernzerhof (PBE) [36]. The Kohn-Sham equations were solved using the plane-wave basis set (PAW) [37]. The cut-off energy was chosen equal to 400 eV. The force tolerance was 0.5 eV/nm and the energy tolerance for the self-consistency loop was 10 −5 eV. The Brillouin zones were sampled sampled using Monkhorst-Pack grids [38] including 3×3×3 k-points. We performed spinpolarized calculations in all cases initializing Fe atoms to have 3.63 µ B , and Pd atoms to be in the paramagnetic state (0 µ B ). The structures are described as consisting of a filled FCC host matrix formed by Pd atoms, with Fe ions substituting octahedrally coordinated sites only (see Fig. 1). The 3×3×3 unit cell parameter a = 1.18146 nm.

Magnetic properties of Pd 1−x Fe x with varying concentration and uniform impurity distribution
At the first stage, various concentrations of Fe solute in a 3×3×3 Pd-supercell with uniform distribution were considered. We increased the number of Fe atoms in the unit supercell from one to eleven by substituting Pd ions. At low Fe + contents (x = 0.01 to 0.03) we obtained a negligible cell magnetization (Fig. 2 a). At such concentrations, Fe solute atoms are non-magnetic (Fig. 2 b) and do not magnetize the surrounding Pd atoms. As the impurity concentration increases, iron becomes magnetic (Fig. 2 b) and magnetizes the surrounding Pd atoms (Fig. 3 a). In particular, about fourteen Pd atoms received significant magnetic moments, as shown in Fig. 1 b. We found that the total magnetization of a cell with seven Fe solute atoms gives rise to the highest total magnetic moment calculated per Fe atom which equals ≈8 µ B (including Fe magnetic moment). This value is in agreement with the experimental results shown also in Fig. 2. After the peak point, the curve slowly decreases in agreement with results of Esmaeili et al. [23] obtained for MBE films, and of Crangle [2] found for the bulk. Our curve is somewhere in the middle of two mentioned results. However, there is a contradiction with experiment at low impurity concentrations. That might be due to the fact that in the calculations were performed assuming 0 K and we got Fe solute atoms, as a result of optimization, to be non-magnetic at low impurity concentrations, whereas experimental measurements were carried out at final temperatures. Besides, the theory of impurity ferromagnetism suggests that at very low impurity concentration, when the distance between impurity ions is large enough, the oscillatory potential prevails over the ferromagnetic one, so that the impurity ferromagnetism does not occur and the spin glass is formed in the alloy [39]. In comparison with assumed theoretical value of x = 10 −4 [39] for the critical Fe content, ab initio gives x = 0.03. At the same time, the calculated magnetic moments for Pd atoms are in a very good agreement with the experimental data obtained by Crangle [2] for bulk alloys (Fig. 3 a). One can notice that the magnetic moment of Pd sharply increases from ≈ 0.05 to 0.35 µ B when the iron concentration exceeds three atoms per 3×3×3 cell. After that, the magnetic moment reaches a plateau with the highest value of the moment per Pd atom equal to ≈ 0.35 µ B , which is consistent with the experimental data shown also in Fig. 3Ṫhe magnetic moment of iron also reaches a plateau with a constant value of ≈3.25 µ B (Fig. 2 b), which is lower than the theoretical maximum, but it is higher than the value of 2.8 µ B obtained in Ref. [2] and close to the value of 3.5 µ B obtained by Neutron diffraction experiments [40,41]. We also checked the concentration dependence of the total energy, which shows a descending character and might mean that the system tends to have Fe ions in the matrix (Fig. 3 b).
The discussed findings confirm the reproducibility of experimental results [2,23] as well as theoretical predictions [39] for Pd 1−x Fe x alloys with the ab initio instrument, which will be used in further discussions regarding the positions of Fe impurities in the host matrix. At the second stage, we consider the influence of Fe-Fe distance in the alloy with Pd 0.98 Fe 0.02 composition if two iron solute atoms reside in the 3×3×3 Pd-supercell. We found that at particular distances (≈ 0.2-0.35 nm) the total magnetization as well as the magnetic moment per Pd atom reach a wide maximum (Fig. 4 a). The highest total magnetization per Fe atom equals ≈ 8 µ B , while the highest magnetic moment of the Pd atom is ≈0.2 µ B (Fig. 4 b). This maximum of magnetization corresponds to the minimum of the total energy (Fig. 4 c). We also tested a different (another concentration) configuration of three Fe solute atoms in the host Pd supercell and obtained similar dependence with a maximum of magnetization at analogous Fe-Fe distances.

Pd
With this finding, we may answer the question mentioned in the introduction section. After implantation, impurities are located inhomogeneously: some Fe solute atoms are too close, others -too distant. However, the annealing process used in experimental investigations leads to a more uniform redistribution of impurities in the host Pd-matrix with lower energy. Obviously, the system tends to have the lowest energy, and this may happen during annealing. This is illustrated in Fig. 5, where three possible iron configurations are discussed. Closely located magnetic impurities (Fig. 5 a) form two overlapping magnetic clusters of neighboring Pd atoms. This situation is energetically unfavorable and the system of Pd 0.98 Fe 0.02 has a relatively low total magnetic moment ≈ 0.5 µ B (Fig. 4 a) calculated per Fe atom. As soon as solute atoms move away from each other (during annealing, for instance) (Fig. 5 b), the total energy lowers reaching a minimum (Fig. 4 c). This energy minimum corresponds to the maximum of magnetization (≈ 8 µ B at 0.2-0.35 nm in Fig. 4 a) and the solute atoms configuration depicted in Fig. 5 b. Finally, distant magnetic clusters (Fig. 5 c) weakly feel each other and may even have opposite magnetic moment directions, which results in low spontaneous magnetization and higher energy.

Conclusion
In the present work, we have demonstrated that calculations within the density functional theory can reproduce basic experimental results obtained for Pd 1−x Fe x alloys, in particular, we have shown that Fe solute in Pd host matrix induce ferromagnetism, which extends to very low concentrations (x = 0.03). The magnitude of the total magnetic moment is very large, about 8 µ B per Fe solute atom and it smoothly decreases with increasing impurity concentration.
We also investigated the dependence of magnetization on the mutual positions of dissolved Fe in the Pd host matrix. The aim was to answer the experimental question about the nature of the increase in T C and M upon annealing. We have demonstrated that the magnitude of the total magnetic moment of Pd 0.98 Fe 0.02 alloy depends on impurity positions, and, as a consequence, the magnetization increase during annealing relates to the impurity redistribution in the Pd matrix.