Alloying effect on the order–disorder transformation in tetragonal FeNi

Tetragonal (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox{L1}}_{0}$$\end{document}L10) FeNi is a promising material for high-performance rare-earth-free permanent magnets. Pure tetragonal FeNi is very difficult to synthesize due to its low chemical order–disorder transition temperature (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx {593}$$\end{document}≈593 K), and thus one must consider alternative non-equilibrium processing routes and alloy design strategies that make the formation of tetragonal FeNi feasible. In this paper, we investigate by density functional theory as implemented in the exact muffin-tin orbitals method whether alloying FeNi with a suitable element can have a positive impact on the phase formation and ordering properties while largely maintaining its attractive intrinsic magnetic properties. We find that small amount of non-magnetic (Al and Ti) or magnetic (Cr and Co) elements increase the order–disorder transition temperature. Adding Mo to the Co-doped system further enhances the ordering temperature while the Curie temperature is decreased only by a few degrees. Our results show that alloying is a viable route to stabilizing the ordered tetragonal phase of FeNi.

From Fig. 2 and Table 1, we can see that only two alloy systems have smaller equilibrium volumes than pure FeNi in the ordered phase: Fe 1−x Cr x Ni and Fe 1−x Co x Ni . Furthermore, we observe that Fe substitution leads to smaller equilibrium volumes compared to Ni substitution. This is due to the stronger decrease of the total magnetic moment when Fe is replaced by a dopant compared to FeNi 1−x M x . More Fe means larger total magnetic moment, and systems with large magnetic moments tend to have large equilibrium volumes due to the magnetic pressure. In addition, we find that Al, Ti, and Mo increase the volume, which is consistent with the large atomic radii of Al, Ti, and Mo as compared to those of Fe and Ni. The volume increase is the largest for the FeNi 1−x Ti x alloys, which is consistent with the above observation. The large volumes of the FeNi 1−x Ti x alloys are accompanied by large c/a ratios. Compared to the undoped FeNi alloy, Cr increases the volumes when doping the Ni layer and decreases them when doping the Fe layer. This doping induced reduction of the volume happens because the atomic radii of Cr and Co are not as big as those of Al and Ti and the decreasing Fe content lowers the total magnetic moment, which in turn decreases the equilibrium volume. Cobalt addition negligibly affects the volume when doping on the Ni layer. There is practically no volume change in Fig. 2 and also changes in the magnetic moments are quite small as Table 2 shows. According to the present data, Co can substitute Ni almost perfectly, which is attributable to the fact that Ni and Co are similar chemically. However, Co decreases the volume of FeNi when doping the Fe site.
Total and atomic spin magnetic moments of M-doped FeNi are shown in Table 2 for the ordered ( η = 1 ) and fully random ( η = 0 ) phases. The bottom rows of both tables show the magnetic moments of undoped FeNi for comparison. The total magnetic moments of M-doped FeNi are naturally decreased when Fe is replaced by the dopant. Likewise, there is a slight reduction in the magnetic moments when Ni gets replaced, except when doping with Co. All dopants except Co show antiferromagnetic coupling, although for Al, Ti and Mo the moments of the dopants are very small.
It should be noted that doping with Cr causes bigger reductions in total magnetic moment than the nonmagnetic Al, Ti, and Mo dopants. Two factors contribute to this. Firstly, the Cr-doped alloys have smaller equilibrium volumes compared to the Al, Ti, and Mo-doped alloys, which means reduced moment due to the magneto-volume effect. Secondly, as Table 2 shows, Cr favors strong antiferromagnetic coupling, which reduces the total magnetic moment.
In our previous paper, we established the order-disorder transition temperature T od of undoped FeNi to be 559 K 15 . Here our goal is to understand in what way alloying affects this temperature, and we are specifically looking for ways to increase this transition temperature. The present theoretical order-disorder transition temperatures  Fig. 3. The order-disorder transition temperature T od changes differently depending on whether we dope the Fe layer or the Ni layer. A general feature we can identify is that in most cases doping the Fe layer leads to higher transition temperatures compared to doping the Ni layer. Only the Cr case is such where doping the Fe layer decreases the transition temperature compared to Ni layer doping. We ascribe this to the strongly antiferromagnetic nature of Cr.
The changes in the equilibrium volume shows interesting correlation with the transition temperature. For Al, Fe 1−x Al x Ni with smaller volume has a larger transition temperature, while FeNi 1−x Al x with bigger volume has a smaller transition temperature. The FeNi 1−x Ti x alloy, which has very large equilibrium volume at high doping www.nature.com/scientificreports/ levels, shows rapid decrease of the transition temperature. The FeNi 1−x Co x alloy on the other hand has small equilibrium volumes, but increased transition temperatures. In all alloy cases, except FeNi 1−x Ti x , the maximum T od occurs around x = 0.05 . We expect the initial increase of T od for small values of x to be caused by the additional configurational entropy created by the dopant. Due to doping the configurational entropy of the ordered state is larger than zero. For small values of x, the proportional entropy gain of the ordered state should be larger than that of the random state. The entropy of the ordered state stabilizes the ordered configuration, as compared to the random state, which leads to the increasing T od for small values of x. Beyond x = 0.05 , T od starts decreasing, because the DFT total energy difference between ordered and random states starts to decrease significantly, as shown in Fig. 4. By inspection of the DFT energy differences   Fig. 3 shows a sizable T od decrease when going from x = 0.05 to x = 0.10 , but this decrease is smaller than that of Fe 1−x Cr x Ni alloys, which exhibit significant energy differences. In Fig. 5, the Curie temperatures T c are presented for different alloy systems at ordered ( η = 1 ) and disordered ( η = 0 ) phases, respectively. All ordered phases have higher T c than the disordered phases. This predicts that the positive magnetic contribution are likely to increase the order-disorder transition temperature T od . Fe 0.90 Co 0.10 Ni alloy has the highest Curie temperatures, which is to say that adding Co into the alloy raises the Curie temperature, which is sensible given the rather high ( ≈ 1400 K) Curie temperature of pure Co. The Curie temperature of L1 0 Fe 0.93 Co 0.05 Mo 0.02 Ni is by only 14.2 K smaller than that of L1 0 FeNi.
The highest theoretical order-disorder transition temperatures predicted in the present study are for Fe 0.95 Co 0.05 Ni (618 K) and Fe 0.93 Co 0.05 Mo 0.02 Ni (630 K). In both cases, it is assumed that Co and Mo replace Fe within the Fe sublattice. In practice, however, the situation could be more complex. Alloying additions could induce Fe and Ni intermixing or phase decomposition. In the following, we study the energetics of two chemical processes which might prevent the formation of the L1 0 phase with Co located on the Fe site in Fe 0.95 Co 0.05 Ni . In Fig. 6 F). We compute the free energy for each of these processes and check the phase stability as a function of temperature. In this study, the free energies include the configurational and vibrational contributions, but neglect the electronic and magnetic contributions.
First, we discuss the Co doping on the Fe site (cases A, B, and C). According to the free energy differences in Fig. 6, at low temperatures the ordered structure (case A) is more stable than the one with interlayer mixing (case B), but less stable than the phase separated case (case C). At elevated temperatures (above ∼ 500 K) these trends are reversed so that the ordered single phase structure is more stable than the phase separation, and the interlayer www.nature.com/scientificreports/ mixing becomes more stable than the ordered structure. Next, we repeat the Co doping for the Ni site (cases D, E, and F). The overall trends are similar to the doping on the Fe site case. Therefore, the thermodynamic stability seems to prevent the formation of the ideal L1 0 Fe 0.95 Co 0.05 Ni or FeNi 0.95 Co 0.05 phases. Considering that phase separation requires substantial diffusion, perhaps its impact is less relevant around the ordering temperature. However, the interlayer mixing, which is mainly driven by the configurational entropy, is predicted to affect the Co partition and thus the largest achievable ordering temperature for the Co-doped FeNi system might be somewhat below the largest values from Fig. 3.

Conclusions
We have studied the effect of alloying on the properties of tetragonal

Methods
The first-principles calculations were performed within the exact-muffin-tin orbitals (EMTO) method 16-18 based on Density Functional Theory 21 . The s, p, d, and f orbitals were included in the EMTO basis sets. The singleelectron Kohn-Shan equations were solved by the Green's function technique and the compositional disorder was treated using the coherent-potential approximation (CPA) 19,20 . The total energies were computed via the full charge density technique 22 . The exchange-correlation functional was approximated by the Perdew, Burke, and Ernzerhof (PBE) 23 generalized gradient approximation. The magnetic transition temperatures were estimated using the UppASD spin dynamics code 24 . The free energies of ordered, partially ordered and disordered Fe 1−x M x Ni and FeNi 1−x M x phases were expressed as a function of η, where E 0K is the internal energy per unit cell at 0 K, S conf is the configurational entropy, F vib , F el and F mag are the vibrational, electronic and magnetic free energies, respectively. According to the static Concentration Waves method 25 , the configurational entropy of L1 0 Fe 1−c (Ni 1−x M x ) c (or ( Fe 1−x M x ) 1−c Ni c ) were described as a function of LRO parameter η in the form Here the atomic fraction of the solute c equals to 0.5 and x is 0.05 or 0.10. Total atomic number N equals to 4 in L1 0 structure. Detailed information about the approach can be found in Ref. 25 .
The vibrational contribution to Helmholtz free energy, F vib (V , T, η, x) = E vib − TS vib , was described by Debye model with the Debye temperatures determined by the tetragonal elastic parameters. The electronic contribution to free energy was estimated by F el ≈ − 1 2 TS el (V , η, x) ≈ − 2π 2 3 k 2 B T 2 N el (ε F , η, x) , where electronic density of state N el (ε F , η) is approximated to be constant in the neighborhood of the Fermi level ǫ F . The magnetic contribution to free energy, F mag (V , T, η, x) = −TS mag (V , T, η, x) = −T i� =j J ij µ i µ jêiêj where J ij is the Heisenberg exchange interaction between atoms i and j, and µ i and µ j are the local magnetic moments on sites i and j. The order-disorder temperature T od was then obtained by computing ∂η/∂T. (1) F(V , T, η, x) = E 0K (V , η, x) − TS conf (η, x) + F vib (V , T, η, x) + F el (V , T, η, x) + F mag (V , T, η, x)