Electronic structure and magnetic properties of the Fe16N2 doped with Ti

Journal of Magnetism and Magnetic Materials 420 (2016) 75–80

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Electronic structure and magnetic properties of the Fe16N2 doped with Ti D. Benea a,n, O. Isnard b,c, V. Pop a a

Babes Bolyai University Faculty of Physics, 400084 Cluj-Napoca, Romania Univ. Grenoble Alpes, Inst. Neel, F-38042 Grenoble, France c CNRS, Inst. NEEL, F-38042 Grenoble, France b

art ic l e i nf o

a b s t r a c t

Article history: Received 29 April 2016 Received in revised form 23 June 2016 Accepted 28 June 2016 Available online 5 July 2016

Detailed theoretical investigations on the electronic and magnetic properties of the α″-Fe16N2 phase doped with Ti have been performed. The investigations done using the Korringa–Kohn–Rostoker (KKR) band structure method are based on the Local Spin Density (LSDA) and LSDA þ U approximations, whilst the disorder in the systems has been accounted for by means of the Coherent Potential Approximation (CPA). We found that the Ti atoms substitute with preference on Fe crystal sites with N atoms as nearest neighbors (8h). In the spin resolved density of states (DOS) the covalent nature of the interatomic bands and the N-2p and Fe/Ti-3d hybridization can be observed. The change in the local environment by Ti substitution is evidenced in the distribution of local magnetic moments and hyperfine magnetic fields. The total magnetic moment of the α″- (Fe1 − xTix )16N2 compound increases by doping, due to the magnetovolume effect. The same increase is observed for the Fe magnetic moments on different crystal sites. The increase of the total magnetic moment by Ti doping cannot outpace the volume increase and, as consequence, the estimated volume magnetization decreases. As the Ti substitution is used in practice to increase the thermal stability of the α″-Fe16N2, our calculations may offer insights into the change of the magnetic properties by doping, which are very important for practical applications. & 2016 Elsevier B.V. All rights reserved.

Keywords: Electronic band structure Magnetic moments Hyperfine fields Density of states

1. Introduction The magnetic properties of Fe nitrides have been studied extensively in the last 40 years, since a huge saturation magnetization has been reported in 1972 by Kim and Takahashi [1] for the Fe–N films. The outstanding magnetic properties in Fe nitrides have been attributed to α″-Fe16N2 phase, with very large magnitude of the magnetic moments (∼3.0 μB) reported by Sugita et al. [2] for a single crystal grown by molecular beam epitaxy (MBE) method. The α″-Fe16N2 phase has been first synthesized in bulk by Jack [3] by quenching of the cubic nitrogen austenite γ-FeN and producing the α′-FeN phase. This intermediate α′-FeN phase orders by a subsequent low temperature annealing to produce the α″-Fe16N2 phase, which is metastable, the decomposition into α-Fe and Fe4N being observed at ∼500 K [3]. The existence of giant magnetic moments in α″-Fe16N2 is still under debate, many experimentalists [4–7] reported magnetization values much lower than the initially reported values [1,2]. The difficulties consist in preparation of the single phase crystals, phase identification and n

Corresponding author. E-mail address: [email protected] (D. Benea).

http://dx.doi.org/10.1016/j.jmmm.2016.06.083 0304-8853/& 2016 Elsevier B.V. All rights reserved.

the determination of the volume fraction of each Fe–N phase. Also, the lack of accurate values for magnetization of other Fe–N phases makes the measurements of the magnetization for the α″-Fe16N2 phase very tedious and imprecise. The interest for this Fe-N phase reappeared in the last years, due to the increase of the rare-earth price and therefore, many new attempts to discover cheaper alternatives for high energy permanent magnets occurred. At present, giant magnetization (∼3.1 μB) of the epitaxial thin film of Fe16N2 has been confirmed [8], but its magnitude depends on the film depth. Theoretical calculations performed using the LSDA by different methods (ASW, LMTO, FLAPW, LCAO) [7,9–18] found an average magnetic moment increased with respect to the bulk iron, between 2.3 and 2.6 μB, but far from the values of 3.0–3.3 μB deduced from the experiment [2,19,20]. The calculations beyond the LSDA, by including the electronic correlations within LSDA þU (Lai et al. [21]) found an increased average magnetic moment of 2.85 μB/Fe. Using the GGA þ U method, Shi et al. [22] showed that by increasing the Coulomb interaction U between 3d electrons of Fe, the average magnetic moment is also increasing and a giant saturation magnetization can be achieved for large Hubbard U values, as demonstrated earlier [23]. On the other hand, Sims et al.

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D. Benea et al. / Journal of Magnetism and Magnetic Materials 420 (2016) 75–80

[24] used different ways to include the electronic correlations (the hybrid functional method, the GW approximation and the GGA þ U method) and obtained an average spin moment of 2.9 μB, 2.6– 2.7 μB, and 2.7 μB, respectively. A similar value for the average spin moment (2.59 μB) has been obtained by Ke at al. [25] using the quasiparticle self-consistent GW approximation. During many years of study of this magnetic phase, doping with various transition metal elements has been performed [6,26– 31] in order to increase the thermal stability of the magnetic phase. Beside the effects in stabilizing the phase, the doping with 3d elements could change the magnetic properties dramatically. The previous studies [28,29] showed that a small Ti addition (up to 5%) can stabilize the doped α″-Fe16N2 phase up to 700 K. Also, Wang et al. [27] showed that in the case of (Fe,Ti)N-films on Si (001) substrate, for Ti content higher than 10%, the α″ phase cannot be formed. For lower Ti content ( ≤ 10%), the saturation magnetization is higher than the value for bulk Fe, with a maximum at about 5% Ti in the films, attributed to the α″ phase. In the present article we study the influence of the Ti doping on the magnetic properties of the α″-Fe16N2 phase, using the spin polarized relativistic Korringa–Kohn–Rostocker (KKR) band structure calculations. As only the phases with small amount of Ti are likely to be formed, we limited our study at the (Fe1 − xTix )16N2 compounds with x ≤ 0.05 in order to investigate the changes of the electronic structure, magnetic moments and hyperfine fields for each Fe sublattice.

2. Computational details The electronic structure of the (Fe1 − xTix )16N2 compounds (x ¼0, 0.03 and 0.05) has been calculated self-consistently by means of the spin polarized fully relativistic Korringa–Kohn–Rostoker (KKR) method in the atomic sphere approximation (ASA) mode [32]. The calculation method is based on the KKR–Green's function formalism that makes use of multiple scattering theory. The local spin density approximation (LSDA) for the exchange-correlation energy using the Vosko, Wilk and Nusair (VWN) parameterization was used [33]. Additional, the strong on-site Coulomb interactions from the localized 3d electrons of Fe/Ti atoms have been accounted by LSDA þ U method. The LSDA þ U implementation [34] allows to deal with disordered systems in combination with Coherent Potential Approximation (CPA) [35,36]. The double-counting is treated by so-called atomic limit expression suggested by Czyczyk and Sawatsky [37]. The effective on-site interaction has been parametrized by the Hubbard U and the Hund exchange interaction parameter J. The values of U and J are sometimes used as fitting parameters and the previous studies performed to describe the α″-Fe16N2 phase by LSDA þ U calculations used very different values of these parameters [21,24,38–40], based on analogy with other systems or by applying a model. In the present study, we used the on-site Coulomb interaction (Hubbard U) between the localized 3d electrons derived by Sims et al. [24] from first principle computation based on the constrained random-phase approximation (cRPA). The Hubbard U parameters are 3.12, 3.52 and 3.99 eV for Fe 4e , 8h and 4d , respectively, while the J parameters values are 0.59, 0.61 and 0.64 eV for the same sequence of sites. We considered for the Ti atoms the same U and J values as for the Fe atoms. The hyperfine field calculations use the fully relativistic expression [41] for the hyperfine interaction operator that describes → the coupling between the electronic current density j = ec→ α and → the vector potential An created by the nuclear dipole → μn :

→ → → Hhf = e→ α ·A n ( r ) = e→ α ·(→ μn × r )A n (r )

(1)

where the radial dependence An (r ) is r  3 for a point nucleus. The contribution of the core electrons and valence band electrons to the hyperfine interaction can be calculated separately [41,42]. The contribution to the Hhf due to the non-s electrons is included in the present calculation. In particular, the inclusion of the spin–orbit coupling makes the valence contribution for a given orbital quantum number l to be proportional to the orbital magnetic moment μorb [41].

3. Results and discussions The α″-Fe16N2 compounds crystallize in a bct structure (I4/ mmm space group) with ordered N atoms at the octahedral interstitial sites [3] (see Fig. 1). In this type of compound, the Fe atoms occupy three inequivalent crystal sites ( 4e , 8h and 4d ), whereas N is located on the 2a crystal site. The structure can be seen as a 2  2  2 supercell of bcc Fe with two additional N atoms at interstitial octahedral positions. The Fe sites forming the octahedron are sitting in the upper/lower corner ( 4e sites) and in the horizontal plane ( 8h sites), respectively. The Fe 4d sites do not have N atoms as nearest neighbors. In general, substitution with a 3d element in Fe–N phases is done with an element which has an increased affinity to N compared with Fe. In this way, the thermal stability of the system increases and the 3d element acts like a trap for N, avoiding its diffusion out of the sample. The choice of Ti as substitute for Fe has impact also on the unit cell of α″-(Fe,Ti)16N2, as the atomic radii of Ti (0.176 nm) is larger than the corresponding value for Fe (0.156 nm) [43], expanding in this way the unit cell. The lattice constants of α″-(Fe0.95Ti0.05)16N2 determined by Wang et al. [29] for thin films (∼50 nm) are a ¼0.613 nm and c ¼0.632 nm, with ∼7% and 0.5% increased, respectively, compared with α″-Fe16N2 reported by Jack [3] (a ¼0.572 nm and c¼0.629 nm). The volume expansion corresponding to these experimental measurements is 15.4% compared with the undoped α″ phase. The increase of the interatomic distances by Ti for Fe substitution can be seen in Table 1. Here we show the changes of the nearest neighbor interatomic distances by Ti addition for each Fe site. The distances between Fe atoms and their first neighbors are increasing by Ti substitution with about 7% for Fe 8h and with 5% for Fe 4d , respectively. Less affected by the Ti substitution are the distances between Fe 4e atoms and their next neighbors. On the other hand, other studies [44] show that the substitution with 3.2% Zr (with higher atomic radius, 0.206 nm compared with 0.176 nm for Ti) produce a volume expansion of only 2% in

Fig. 1. The structure of the Fe16N2 compound.

D. Benea et al. / Journal of Magnetism and Magnetic Materials 420 (2016) 75–80

Table 1 (Fe,Ti)16N2 : Nearest neighbor distances of the Fe sites in the Fe16N2 and (Fe0.95Ti0.05)16N2 compounds (for experimental lattice parameters [3,29]). (Fe0.95Ti0.05)16N2

1 1 8

Fe 4e Fe 8h Fe 4d

Distance (Å)

N N

Distance (Å)

1.825 1.98 2.56

Fe 8h

1.83 2.12 2.68

Sites

a¼ 0.572 nm

a¼ 0.613 nm

4e

c¼ 0.629 nm

c¼ 0.632 nm

ΔEtot (mRy.)

ΔEtot (mRy.)

40.258 67.431 25.943 42.547 58.993 0 89.747

36.640 69.407 36.150 38.596 69.580 0 73.489

4d

Ti content (%) 5 10 0 10 0 0 20

5 0 5 5 0 10 0

5 10 10 0 20 0 0

Method

δV /V0

x¼0

LSDA

0

LSDA þ U Exp. [7]

N

Fe 4e

Fe 8h

ms ml ms ml ms

 0.18 0.00  0.19 0.00

2.17 0.07 2.48 0.11 2.33

2.33 0.06 2.53 0.09 2.45

Ti 8h

Fe 4d 2.75 0.07 2.98 0.11 3.05

x¼3

LSDA þ U

1.9

ms ml

 0.18 0.0

2.47 0.12

2.57 0.09

 0.77 0.03

2.94 0.12

x¼5

LSDA þ U

1.9

ms ml

 0.16 0.0

2.44 0.11

2.57 0.09

 0.75 0.03

2.90 0.12

x¼5

LSDA þ U

15.4a

ms ml

 0.20 0.00

2.66 0.11

2.88 0.09

 0.90 0.02

3.02 0.13

a

Ti x = 0 Ti x = 5 Ti x =3

2.7

1.95 1.9

Ti x = 0 Ti x = 5 Ti x = 3

1.85 1.8

bcc Fe

1.75 1.7

0

5

δ V/ Vo(%)

10

15

Fig. 2. (a) The average Fe magnetic moment 〈μFe 〉 (in μB) vs. the volume expansion (%) in the (Fe1 − xTix )16N2 compound. (b) The volume magnetization Mvol vs. the volume expansion (%) in the (Fe1 − xTix )16N2 compound.

Table 3 Spin (ms) and orbital (ml) magnetic moments (in μB) in the α″- (Fe1 − xTix )16N2 compounds (x ¼0, 0.03 and 0.05), calculated by the KKR-ASA using the LSDA and the LSDA þ U approach. Ti content

2.8 2.75

2.62

Table 2 α″-(Fe0.95Ti0.05)16N2: The total energy variation by LDA þ U calculations for different distribution of Ti atoms on Fe crystal sites ( 4e , 8h and 4d ).

8h

2.85

2.65

-6

Type

Mvol x 10 (A/m)

Nr.

2.9 < μ Fe> ( μ B)

Fe16N2

Atom

77

Lattice constants according to Wang et al. [29].

Fe–Zr–N films with nitrogen content of 11 at% (corresponding to the α″-phase). In this sense, the 15.4% increase of the volume determined by Wang for α″-(Fe0.95Ti0.05)16N2 could be influenced by the details of the growing process. In our study for α″(Fe1 − xTix )16N2, a gradual increase of the a lattice constant will be considered, from a¼ 0.572 nm to 0.613 nm with the constant value of c ¼0.632 nm. As consequence, the volume range between the initial cell volume V0 [3] of the undoped compound and the Vmax, increased with 15.4% [29] of the α″-(Fe0.95Ti0.05)16N2. In the case of α″-(Fe0.97Ti0.03)16N2, a smaller volume expansion (5%) will be considered. The electronic and magnetic properties of the doped α″-phase will be discussed considering the volume expansion δV /V0 as parameter. Band structure calculations have been performed for the α″-(Fe 0.95Ti 0.05)16N2 compound in order to determine the preferential occupation of Ti on Fe inequivalent crystal sites ( 4e , 8h and 4d ). Total energy calculations have been performed (see Table 2), taking into account different occupation of the Ti atoms on the Fe sites. As deduced from the total energy calculations, the Ti atoms

have the tendency to occupy the crystal sites with the N atoms as first neighbors ( 8h sites). Much larger affinity of N for Ti than Fe plays in favor of the tendency to form direct bonds between Ti and N [45]. This preferential occupation will be considered in the further calculations. The increased magnetic moment reported in the α″-Fe16N2 phase [2] is related to the expansion of the lattice parameter of bcc Fe, by adding N atoms which enters in an interstitial site of the lattice. The Fe–Fe interatomic distances increase and the high momentum Fe state is prevailing [7]. By Ti doping, the expansion of the lattice constants in the α″-(Fe,Ti)16N2 compounds is higher, but there are also different changes in the environment of the Fe sites, which can influence the magnetic properties. The calculated magnetic moments of Fe in the α″- (Fe1 − xTix )16N2 (x ¼0, 0.03 and 0.05) phases are shown in Table 3 together with the magnetic moments obtained from measurements [7]. As can be seen, for Fe16N2, the LSDA þU approach enhance the agreement with the experimental data. Also, the magnetic moments of the Fe atoms depend strongly on the local environment discussed before. For small volume expansion (1.9%), we notice the opposite influence of the Ti doping on the Fe magnetic moments substituted by Ti ( 8h) and its nearest neighbors ( 4e and 4d sites). By considering the lattice expansion, as the experimental measurements show [27– 29], the magnetic moments on all Fe sites are increasing by doping, due to the magneto-volume effect. Both KKR–LSDA and LSDA þU calculations are in agreement with the experimental measurements [7] finding the highest value of the magnetic moments on the Fe 4d site, whilst the smallest value is attributed to the Fe on 4e site. This sequence of the magnetic moments magnitude is preserved by Ti for Fe substitution. Also, the sequence of the magnetic moments magnitude is similar with other calculations. Such hierarchy of the magnetic moment magnitudes for the different Fe positions has been discussed in terms of influence of local atomic environment by Isnard et al. [46]. The existence of strongly enhanced magnetic moment on a certain Fe position, coexisting with lower magnetic moments on other Fe positions has been also evidenced in other Fe rich intermetallic compound [47]. Using the calculated value of the magnetic moment per formula unit by LSDA þU calculation, the magnetization obtained for the α″-Fe16N2 phase (1.94  106 A/m) is in good agreement with the experiment (2.0  106 A/m) [1], being higher than the

↑ ↓

↑

↑

↓

↓

↑

↑ ↓ ↑

↓

D. Benea et al. / Journal of Magnetism and Magnetic Materials 420 (2016) 75–80

↓

78

Fig. 3. The site-dependent DOS calculated by KKR LSDA þU method for Fe (on 4e, 8h and 4d sites), Ti on 8h site and N, together with the total DOS for the Fe16N2 (filled curves) and the (Fe0.95Ti0.05)16N2 compounds. The origin of the energy scale is the Fermi level.

corresponding value of bcc Fe (1.75  106 A/m). Also, the LSDA þ U calculated average magnetic moment of Fe in α″-Fe16N2 phase (2.7 μB) is in good agreement with other theoretical calculations

[24,25]. The LSDA þU calculated average magnetic moments of Fe and respectively the magnetization as a function of the volume expansion for (Fe0.95Ti0.05)16N2 are presented in Fig. 2.

D. Benea et al. / Journal of Magnetism and Magnetic Materials 420 (2016) 75–80

Table 4 Hyperfine fields (in kGs) in the Fe16N2 compounds. The theoretical hyperfine fields have been decomposed into the core, valence and total (valence þ core) contributions. Ti content

Method

Fe (4e)

Fe(8h)

Fe(4d)

Fe16N2 LSDA

Core Valence Valence þ core

238 4 234

253  57 196

300  25 275

Fe16N2 LSDA þ U

Core Valence Valence þ core

272  33 239

277  50 227

329  24 305

Valence þ core

296 230

316 220

399 310

Fe16N2 Exp. [48] Theor.(rel) [24]

350

(kGs)

200

Bhf

core

v+c

250

, Bhf

300

150

B

(4e)

B

(8h)

B

(4d)

B

(4e)

B

(8h)

B

(4d)

100

-60

Bhf

val

(kGs)

-50

-70

B (4e) B (8h) B (4d)

-80

0

5

δ V/V0 (%)

1015

Fig. 4. The hyperfine fields (core, valence and valence þcore contributions) vs. the volume expansion, calculated by the KKR LSDA þU method for the (Fe0.95Ti0.05)16N2 compound.

The average magnetic moments of Fe increase by expanding the lattice of the (Fe0.95Ti0.05)16N2 compound due to the magnetovolume effect. On the other hand, the decrease of the magnetization with the expansion of the lattice can be seen in Fig. 2, as the enhancement of the magnetic moments cannot outpace the increase in volume. The decrease of the magnetization with volume expansion is more pronounced by increasing the Ti content. The spin-resolved densities of states for the Fe16N2 and for the (Fe0.95Ti0.05)16N2 compounds calculated by the KKR LSDA þ U band structure method using the experimental lattice constants are shown in Fig. 3. As can be seen in Fig. 3, the DOS of Fe16N2 keeps the characteristic features described before [7,16], showing a spindown DOS with two main peaks, having a deep valley between them and the Fermi level EF position being pinned in the middle of this valley. Also, the majority spin-band is almost fully occupied, which is a characteristic of the strong ferromagnetic compounds. On the other side, the DOS of the three inequivalent Fe sites are different in shape, showing different hybridization degrees with

79

the N atoms, visible in the region  13 to 15 eV for Fe 4e and Fe 8h. In particular, these features are related to the N-2p and Fe-3d hybridization for the Fe 4e and Fe 8h sites having the N atoms as first neighbors, but absent for Fe 4d sites, which is more distant from the N atoms. In the case of Ti doped compound, less 3d electrons are accommodated in the valence band, inducing a Fermi level shift. Due to this shift, the spin-down DOS significantly increases at EF, as the Fermi level position is moved from the middle of the valley between the main two peaks towards the lower energy peak. The spin-up DOS at EF is less affected by the Fermi level shift, the spinup channel being fully occupied as before Ti doping. The hyperfine fields calculated using the KKR formalism for Fe16N2 compound are presented in Table 4 together with the results of Mössbauer measurements for Fe16N2 [48]. The contribution of the core and valence electrons at the hyperfine fields of Fe on different sites are evidenced. The main contribution to the hyperfine fields comes from the core electrons, the sequence of the core contributions to the hyperfine fields being proportional with the spin moments: Bhfcore(4d ) > Bhfcore(8h) > Bhfcore(4e ). As can be seen in Table 4, the KKR calculations and the Mössbauer experiments are in disagreement with respect of the hyperfine fields magnitude sequence. On the other hand, there is a good agreement between the hyperfine fields obtained by KKR formalism and the hyperfine fields calculated with other methods [24,48]. The discrepancies between calculated and experimental hyperfine fields can be ascribed to problems dealing with the core polarization contribution when the LSDA is used [13,41,49]. As it was shown in a previous work [50], the contact hyperfine fields for 3d ferromagnets (Fe, Co, Ni and some Fe compounds) are underestimated by about 20–30% by LSDA calculations. As can be seen in Table 4, these discrepancies are slightly corrected by LSDA þU, but the calculated hyperfine fields still have significantly smaller magnitudes compared with the experimental values. Also, a large negative contribution of the calculated Bhfval on the Fe 8h site is changing the magnitude sequence of Bhfv+ c in the theoretical calculations compared with the experimental measurements. The hyperfine fields Bhf evolution vs. the volume expansion by KKR LSDA þU calculations for the (Fe0.95Ti0.05)16N2 compound is shown in Fig. 4 for all Fe sites. The Bhfcore increases with the volume expansion for all Fe sites, whilst the valence contributions Bhfval are decreasing for Fe 4e and 4d sites. For the Fe 8h sites, where Ti atoms are substituting, the valence contribution has a nonmonotonous behavior. The total hyperfine fields Bhfv+ c are increasing for the 8h and 4e sites, being almost independent on the volume expansion for the Fe 4d site.

4. Conclusions The investigations of the electronic and magnetic properties of the α″-Fe16N2 compound doped with Ti show that the Ti atoms have the tendency to occupy the Fe sites around the N site ( 8h sites). This preference, deduced from total energy calculations, is explained by the increased affinity of N for Ti than for Fe. In this way, the direct bonds between Ti and N are favored. The total magnetic moment of the α″- (Fe1 − xTix )16N2 compound (with x¼ 0.03 and 0.05), as well as the magnetic moments on all Fe sites of these compounds increase by doping, due to the magnetovolume effect induced by the lattice expansion [29]. The Fe 4d site is found to carry the highest magnetic moment, in agreement with the experiments [7]. The increase of the total magnetic moment by Ti doping cannot outpace the volume increase and, as consequence, the estimated volume magnetization decreases from 1.94  106 A/m for Fe16N2 to

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D. Benea et al. / Journal of Magnetism and Magnetic Materials 420 (2016) 75–80

1.71  106 A/m for (Fe0.95Ti0.05)16N2. We conclude that despite the enhanced thermal stability reported by experiments [27–30], the Ti doping would be detrimental for the magnetic properties of the α″- (Fe1 − xTix)16N2 phase. The hyperfine fields fully relativistic LSDA þU calculations show an increase of the core hyperfine fields of each Fe crystal site by Ti for Fe substitution. On the other hand, the behavior of the valence electrons contribution (Bhfval ) vs. the volume expansion is different on Fe sites (decrease for Fe 4e and 4d sites and nonmonotonous for Fe 8h site). Therefore, the total Bhf increase with volume for Fe 8h and 4e sites and is almost volume independent for the Fe 4d site.

Novelty We report on the electronic and magnetic properties of the Fe16N2 doped with Ti. Previous experimental studies showed an increase of the magnetization by doping in the range 0–10% Ti, attributed to the α″-phase. We performed band structure KKR–CPA calculations using the LDA þ U method for 5% Ti doping. We report on the preferential site occupation of Ti atoms, the change of the average Fe magnetic moment, the density of states and the hyperfine fields by Ti doping. In contradiction with the experimental studies, we conclude that Ti doping is not appropriate for improving the magnetic properties of the α″-Fe16N2, despite its benefits in increasing the thermal stability.

Acknowledgments The financial support of the UEFISCDI through the grants PN-II-PT-PCCA-2013-4-0971 and PN-II-RU-TE-2014-4-0009 is acknowledged.

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