Journal of Alloys and Compounds 414 (2006) 36–41

Distinct magnetic states of metastable fcc structured Fe and Fe–Cu alloys studied by ab initio calculations L.T. Kong, B.X. Liu ∗ Advanced Materials Laboratory, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Received 1 June 2005; received in revised form 14 July 2005; accepted 18 July 2005 Available online 6 September 2005

Abstract Based on the projector augmented wave method within the generalized gradient approximation, ab initio calculations predict five distinct magnetic states of Fe in fcc structure depending sensitively on their atomic volumes, i.e., with increasing the atomic volume, there appears firstly a complicated coexistence of a non-magnetic, two antiferromagnetic, and a low spin ferromagnetic states, and then a high spin ferromagnetic state. Moreover, the present calculations also predict that the high spin ferromagnetic state of Fe atoms could be stabilized in some nonequilibrium Fe–Cu alloys over a relatively broad composition range, suggesting a possible approach to synthesize new Fe-containing alloys of high performance in magnetization. An intensive discussion is accompanied to compare the predictions with those from experiments as well as from those calculated results reported previously. It turns out that the present calculations could give reasonable interpretation to the richness and/or the complexity of the magnetic behaviors of fcc Fe and the Fe–Cu alloys. © 2005 Elsevier B.V. All rights reserved. PACS: 75.30.Cr; 75.50.Cc; 61.66.Dk; 03.67.Lx Keywords: Metastable FCC-Fe; Fe–Cu compounds; Magnetic property; Ab initio calculation

1. Introduction Among the ferromagnetic transition metals (Fe, Co, Ni), Fe has the highest moment, which makes the Fe containing materials useful in hi-tech applications requiring high magnetization. Meanwhile, due to its not fully filled d band, the magnetic moment of Fe in an alloy is changeable and varies in a broad range from 0 to 3 µB , depending on the structure and composition of the materials. The magnetic properties of Fe have thus attracted great attention, especially the magnetic property of the metastable fcc Fe and fcc structured Fe-containing alloys. In this respect, however, the results obtained so far from various investigations show some complexity and sometimes are even controversy. For instance, it has been reported that fcc Fe could be anti-ferromagnetic (AFM) and/or ferromagnetic (FM) and that the magnetic moments per Fe atom were different while Fe atoms locating ∗

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at different atomic sites in thin films or in a reconstructed surface, etc. [1–3]. Besides, neutron scattering, M¨ossbauer and other magnetic measurements have also revealed the richness of the magnetic orderings of Fe [4]. The richness in magnetic states and close competition between different magnetic states of fcc Fe have been confirmed by various first principles total energy calculations [5–9]. In general, the previous calculations predicted that the non-magnetic (NM) or paramagnetic, FM, and AFM states were quite close in energy and that the relative stability of a specific state depends sensitively on the atomic volume of the Fe atom in the state. For instance, some of the previous calculations predicted an AFM ground state at a small atomic volume, while a high-spin ferromagnetic (HSFM) ground state at a large atomic volume. Recently, a new first-principles calculation approach named the projector augmented wave method (PAW) was proposed by Bl¨ochl [10] and is expected to be able to handle even the most difficult issues with improved precision, especially in treating with the transition metals [11]. We therefore employ the PAW method, in the present study, to carry out a series of

L.T. Kong, B.X. Liu / Journal of Alloys and Compounds 414 (2006) 36–41

calculations for further clarifying the structural stability and magnetic orderings of metastable fcc Fe. In spite of some difference in the calculation methods, almost all the calculations suggested that the fcc structured Fe was the most important/interesting issue, because it could possess a highest magnetic moment per atom among all the equilibrium/non-equilibrium states of the 3d ferromagnetic elements, which, to authors’ view, could be an important theoretical guidance for designing and fabricating new materials of high magnetic performance. In this sense, there are two important aspects for further studies. First is to soundly confirm the existence of the HSFM state by experiments, which, from a philosophical point of view, is necessary for a complete recognition of the state. Second is to find out if some non-magnetic fcc structured elements, e.g. the noble metals of Cu, Ag and Au, could form some alloys with Fe to help in stabilizing the Fe in the HSFM state, as fabrication of pure and bulk fcc Fe is almost impossible. Consequently, investigating the magnetic properties of Fe-noble metal alloys is of vital importance. In this regard, however, the previous studies have only been concentrated on the structural and magnetic properties of Fe grown on the Cu substrates or in the Cu matrices and/or non-equilibrium Fe–Cu solid solutions, yet little attention has ever been paid to the possible non-equilibrium Fe–Cu intermetallic compounds, in which Cu alloying might play an indispensable role in stabilizing the HSFM state of fcc Fe. Accordingly, calculation of the structural stability and the associated magnetic property of the possible non-equilibrium Fe–Cu intermetallic compounds is another important objective of the present study with an aim to provide some guidance for developing new materials of high performance.

2. Calculation method The calculations were carried out by using the Vienna ab initio simulation package VASP [12–14] based on the projector augmented wave pseudopotential [10,11]. In the present study, the 3d7 4s1 state is treated as the valence for the PAW pseudopotential of Fe. The exchange and correlation effects were described by the functionals proposed by Perdew and Zunger [15], with generalized gradient corrections [16]. Brillouin-zone integrations were performed according to the Monkhorst–Pack scheme [17]. For calculations of pure Fe, a plane-wave energy cutoff of 334.9 eV was used, and while treating the Fe–Cu alloys, a cutoff of 341.6 eV was employed. Besides, for all spin polarized calculations, the Vosko–Wilk–Nusair interpolations were used for the correlation part of the exchange correlation functional [18].

3. Magnetic orderings of fcc Fe The energy–volume correlation of bcc Fe is first calculated and then fitted to the Murnaghan Equation of State

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Fig. 1. Correlations of total energies (lower panel) and magnetic moments (upper panel) against the lattice constant of fcc iron obtained by ab initio calculations within the GGA.

[19] to obtain its cohesive property as well as to certify the validity of the present calculation method. We find a FM ground state for the bcc Fe, and its lattice constant, magnetic moment, and bulk modulus are calculated to be ˚ 2.21 µB and 1.76 Mbar, respectively, which are in 2.83 A, inspiringly good agreement with the experimental data of ˚ [20], 2.22 µB [21], and 1.72 Mbar [21], respectively, 2.87 A as well as with those calculated by Kresse et al. who treated 3d6.5 4s1.5 as valence for the PAW pseudopotential of Fe [11]. The above agreements certify the validity of the present calculation method, based on which further study is pursued to investigate the magnetic orderings of fcc Fe. Fig. 1 shows the calculated correlations of the total energy and the corresponding magnetic moment versus the lattice constant for fcc Fe with different magnetic orderings. The fitted cohesive properties and the magnetic moments are listed in Table 1, together with some values cited from the available experimental results and/or from the previous calculations. As can be seen, firstly, all the fcc phases of Fe have a positive heat of formation with respect to the bcc FM ground state of Fe, which is consistent with the well-known fact that the FM bcc phase is the equilibrium state of Fe at low temperatures. Secondly, the predicted cohesive properties of different fcc states are quite compatible with the experimental results and/or the previous calculations. Thirdly, the fcc Fe shows rather complicated magnetic orderings, depending sensitively on its spacing distance.

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L.T. Kong, B.X. Liu / Journal of Alloys and Compounds 414 (2006) 36–41

Table 1 Calculated minimum total energy (Emin ), heat of formation (
H), lattice constant (a), equilibrium atomic volume (V0 ), bulk modulus (B) and magnetic moment (MM) of fcc iron with different magnetic ordering Emin (eV/atom)


H (eV/atom)

˚ a (A)

˚ 3 /atom) V0 (A

B (Mbar)

MM (µB )

NM

This work Other work

−8.076

0.140

3.448 3.457 [7]

10.245

2.81 2.93 [7]

0 0

LSFM

This work Other work

−8.078

0.139

3.464 3.488 [7]

10.390

1.61

0.62 1.02 [7]

HSFM

This work Other work

−8.060

0.156

3.632 3.636 [7] 3.645 [21]

11.981

1.68 1.71 [7]

2.54 2.57 [7]

AFM-1

This work Other work

−8.077

0.139

3.452

10.283

2.49

0.46 0.50 [22,23]

AFM-2

This work Other work

−8.075

0.141

3.471 3.496 [7] 3.562 [21]

10.457

1.88 1.93 [7]

0.72 1.30 [7] 0.70 [24,25]

First, comparing with some of the previous calculations, which predicted an AFM ground state [5,7], the present calculations predict a FM ground state at a lattice constant of ˚ for fcc Fe. Second, the present calculations locate a < 3.57 A two different AFM states, instead of one predicted by the previous calculations. One sees clearly from Fig. 1 that when the ˚ there lattice constant of fcc Fe lies below about a = 3.57 A, coexist a NM, a low-spin ferromagnetic (LSFM), and two ˚ AFM states. The NM phase equilibrates at about a = 3.448 A with a minimum total energy of −8.076 eV/atom, while the LSFM one has its minimum total energy of −8.078 eV/atom ˚ The magnetic moment of the LSFM fcc at about a = 3.464 A. Fe at its equilibrium atomic volume was calculated to be about 0.62 µB . Regarding the AFM state, the usual AFM configuration with alternating layers of ␣ and ␤ spins along the [0 0 1] crystalline direction was adopted during the calculations. The calculated results suggest that its total energy actually has two ˚ with a minminima. The first one locates at about a = 3.452 A, imum total energy of about −8.0767 eV/atom, which could be denoted as AFM-1. When its lattice constant expands ˚ a phase transition occurs and the fcc Fe to about 3.467 A, transforms into a second AFM state (AFM-2), which equili˚ with a minimum total energy of brates at about a = 3.471 A −8.075 eV/atom. In spite of a same spin configuration, the equilibrium magnetic moments of the two AFM states are different, i.e., 0.46 µB for AFM-1 and 0.72 µB for AFM-2, respectively. Besides, the fitted bulk moduli of the two AFM states are also quite different from each other. One sees from Table 1 that the bulk moduli of the AFM-1 state is about 2.49 Mbar, while it is only about 1.88 Mbar for the AFM-2 state. These differences in the equilibrium lattice constants, minimum total energies, bulk moduli and magnetic moments clearly suggest that the two AFM states may essentially be the different states of fcc Fe. Incidentally, it is generally understood that the AFM ordering lowers the symmetry on the lattice and hence results in room for structural relaxation, which, in turn, might further lower the energy. In the present case, however, the calculations suggest that it does not have

a prominent effect on the fcc Fe. When the lattice constant of ˚ there appears another the fcc Fe increases up to above 3.57 A, ˚ with a magnetic state, which equilibrates at about a = 3.632 A magnetic moment of about 2.54 µB , and this state is usually denoted as a high-spin ferromagnetic (HSFM) state of fcc Fe. Another interesting finding from Fig. 1 is that for each magnetic state (AFM-1, AFM-2, LSFM, and HSFM), the magnetic moment of the Fe atom in the respective fcc structures seems to be enhanced while the interatomic distance (or the spacing distance) expands. Such a trend was also observed in another ferromagnetic metal Co [26] and was consistent with the corresponding experimental observations [27]. It is also noted that the present calculations show that the minimum total energies of the NM, LSFM, AFM-1 and AFM-2 states are very close to each other, reflecting a close competition among these possible states. These calculated results are in good agreement with the previous calculations as well as with the reported experimental observations. In experiments, the predicted magnetic states of fcc Fe have ever been observed and here we discuss some examples reported in the literature. First, Haneda et al. obtained a nanometer-size Fe particle in fcc structure, which was paramagnetic down to a temperature of 1.8 K and was stable for more than 3 years [28]. This might be the unwonted report of paramagnetic fcc Fe at low temperatures, however, it should lend some support to the predicted non-magnetic state of the fcc Fe by the present ab initio calculations. Second, Bianco et al. obtained an fcc structured Fe phase with ˚ [29], which lies in the spacing distance a = 3.51 ± 0.05 A range of the LSFM state indicated by the present calculation. The magnetic moment of the fcc Fe was found to be lower than that of the bulk Fe, agreeing with the prediction of a LSFM state in the present calculation. Third, Macedo et al. ˚ acquired an fcc Fe with a lattice constant of about a = 3.65 A and found that its magnetic moment was about 2.5(1) µB [30], which is in excellent agreement with the present pre˚ Fourth, dicted value of 2.56 µB for the fcc Fe at a = 3.65 A. a low spin of about 0.5 µB has indeed been observed in the

L.T. Kong, B.X. Liu / Journal of Alloys and Compounds 414 (2006) 36–41

Fe–Ni alloys and it might correspond to the presently predicted AFM-1 state [22]. Besides, some other AFM states ˚ [21,31] and a magnetic moment observed at about a = 3.56 A of about 0.70 µB [24,25] could be correlated to the AFM-2 state predicted by the present calculation. Incidentally, the AFM-I state predicted by Herper et al. [7] might be identical to the AFM-2 state predicted by the present calculation, as the cohesive properties, especially the calculated bulk moduli are quite compatible with each other. The predicted magnetic orderings of fcc Fe may well explain the magnetic behaviors observed in thin Fe films grown on Cu(1 0 0) [31–33]. The Fe films grown at room temperature on Cu(1 0 0) were found to exhibit three distinct phases: (1) a FM fcc structure, when the thickness of the films was less than five monolayers (ML), (2) an AFM fcc (or fct) structure with a surface magnetic live layer, when the thickness of the films was from 5 to 11 ML, and (3) a FM bcc structure, when the thickness of the films was thicker than 11 ML. In particular, when the thickness was between 5 and 11 ML, a quantitative analysis by low-energy electron diffraction (LEED) revealed an enlarged atomic volume of ˚ 3 at the Fe film surface, whereas the interior of the 12.1 A ˚ 3 [31,33]. AccordFe films had an atomic volume of 11.4 A ing to the present calculations, it is believed that for the Fe film with a thickness less than 5 ML, the Fe lattice could be constrained by the Cu(1 0 0) substrate to be of fcc structure, which has a large enough lattice constant falling in the HSFM range of fcc Fe, and therefore shows a FM coupling. With increasing of the thickness, the Fe films tend to relax into a state of lower total energy. As the HSFM state with a large atomic volume has a relatively higher energy than the other states with small atomic volumes, the interior of the Fe film inclines to shrink and falls into the LSFM and AFM ranges. Due to the existence of other influencing factors, such as the reconstruction of the surface, deformation of the lattice, the growth temperature, etc., the interior of the Fe film adopts an AFM configuration. However, because of the surface effect, the surface layer remains a large atomic volume/spacing distance, and therefore still favors an FM coupling. When the thickness of the Fe films was greater than 11 ML, however, the large energy difference between the fcc and the ground bcc FM states would certainly drive the films to transform the fcc state into the equilibrium bcc state. It should be noted that only the collinear spin configurations were considered in the present calculations. Among the various possible spin configurations, there could also exist the non-collinear spin configurations for fcc Fe. For example, Tsunoda has reported a spiral structure with a spiral vector of q = 2π a (0.1, 0, 1) for fcc Fe [4,34]. Besides, by means of modified augmented spherical wave method, Kn¨opfle et al. have calculated the spiral magnetic properties of fcc Fe and found that the spiral structures could have lower total energies than the collinear ones [35]. Because of the complexity of the magnetic ordering, all the metastable fcc Fe, which correspond to those possible non-equilibrium states, are possible to appear under the suitable conditions. As a result, the

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metastable fcc Fe could possess any or any combination of all the possible magnetic orderings, implying that whether a specific magnetic configuration could exist or not was not only determined by its relative stability but also by its encountered condition. Generally speaking, the collinear magnetic states were frequently observed in experiments, although the spiral magnetic state might be energetically more favored than the collinear ones. It is thus expected that if the effect of spin spiral could also be included in the calculation, the precision of the calculation would be improved. One may notice that the calculated HSFM and LSFM states of fcc Fe show a reverse energy sequence comparing to that reported by Kn¨opfle et al. [35], who predicted that the HSFM had a lower minimum total energy than the LSFM one. Such a discrepancy might be originated from the different calculation methods used as well as the different exchange-correlation functionals employed. However, it is believed that the present calculations are relevant, as the predicted energy sequence is consistent with the previous LAPW calculations using different exchange-correlation functionals [7,8], and is in agreement with the fact that the LSFM state has been observed in experiments more frequently than the HSFM state.

4. Structural stability and magnetic property of the metastable Fe–Cu alloys The above calculation results clearly confirmed the existence of the HSFM state of fcc Fe, which possesses a magnetic moment much greater than that of the equilibrium bcc Fe. From an application point of view, the HSFM state of fcc Fe is of great interest. In the past decade, efforts have been made to obtain fcc Fe in its HSFM state. Nonetheless, neither the epitaxially grown thin fcc Fe films nor the Fe-containing precipitates dispersed in some alloys [31–33,36] could readily stabilize the HSFM state of fcc Fe or have a large enough size to vender significance of potential application. An alternative way may then be to stabilize such a HSFM state of Fe in some non-equilibrium intermetallic compounds or solid solutions with a large enough lattice constant. At the same time, the alloying element is required not to considerably degrade the magnetic property of Fe. In this respect, Cu, Ag and Au are appropriate candidates as alloying elements, as the lattice constants of Cu, Ag and Au are favored and some previous experimental observations confirmed the possibility of obtaining some alloys in the Fe–Cu, Fe–Ag and Fe–Au systems. Here, we investigate the Fe–Cu system by performing similar first principles calculations to examine the structural stability as well as the magnetic property of the metastable Fe–Cu alloys. In the calculations, five compositions (Fe3 Cu, Fe2 Cu, FeCu, FeCu2 , and FeCu3 ) and at each composition, some relatively simple structures were considered. For the details concerning the choosing of the compositions and structures, the readers are referred to some of our recent pub-

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L.T. Kong, B.X. Liu / Journal of Alloys and Compounds 414 (2006) 36–41

Table 2 Fitted cohesive properties and magnetic moment per Fe atom of Fe–Cu alloys with different structures at various compositions Phase

a

Fe3 Cu A15 D019 D022 D03 L12 L60

4.57 5.12 3.63 5.74 3.61 3.79

Fe2 Cu C1 C15 Ch

5.54 6.63 3.98

FeCu ␣-NiAs B1 B2 B3 L10 L2a

3.61 4.84 2.89 5.32 3.63 2.99

FeCu2 C1 C15 Ch

5.60 6.70 4.08

FeCu3 A15 D019 D022 D03 L12 L60

4.64 5.16 3.51 5.79 3.65 3.83

V0

Emin


H

B

MM

0.866

11.97 11.85 11.83 11.84 11.71 11.81

−6.920 −6.932 −6.956 −6.941 −6.962 −6.962

0.177 0.165 0.140 0.156 0.135 0.134

1.65 1.56 1.55 1.61 1.30 1.77

2.56 2.49 2.49 2.52 2.43 2.49

1.55

14.18 12.12 14.05

−6.038 −6.451 −6.017

0.685 0.272 0.706

1.18 1.50 1.48

2.79 2.12 0.06

1.00 0.909

14.12 14.13 12.01 18.82 11.95 12.13

−5.240 −5.199 −5.711 −4.536 −5.814 −5.716

0.737 0.778 0.266 1.441 0.163 0.261

1.13 1.11 1.56 0.64 1.55 1.45

2.87 3.03 2.53 3.03 2.62 2.71

1.72

14.60 12.55 16.80

−4.594 −4.918 −4.367

0.636 0.313 0.864

1.01 1.32 0.79

2.90 2.73 3.15

12.47 12.09 12.08 12.11 12.15 12.15

−4.581 −4.674 −4.697 −4.680 −4.630 −4.631

0.276 0.183 0.160 0.178 0.227 0.227

1.34 1.46 1.42 1.42 1.40 1.39

3.08 2.68 2.80 2.79 2.64 2.65

c/a

0.82 1.98

1.63 2.24

0.86

˚ c/a: c/a ratio; V0 : average atomic volume at the equia: lattice constant (A); ˚ 3 /atom); Emin : minimum total energy (eV/atom);
H: heat librium point (A of formation (eV/atom); B: bulk modulus (Mbar); MM: magnetic moment per Fe atom (µB ).

lications [37,38]. During the calculations, the c/a ratios of the D019 , D022 , L60 , Ch and L2a structures are fully relaxed, while that of L10 is kept to be 1. The energy–volume correlations of the phases were thus obtained and their respective fitted cohesive properties are compiled in Table 2. Inspecting Table 2, one sees that all the considered phases have a positive heat of formation, which is in accordance with the immiscible characteristics of the Fe–Cu system. Besides, the bulk moduli of all the Fe–Cu phases are smaller than that of the HSFM fcc Fe, probably due to the introduction of the alloying element of Cu. Concerning the structural stability or energetic sequence of the considered phases, one sees that at the composition of Fe3 Cu, the L12 and the L60 structures are calculated to have nearly the same lowest heat of formation, and possess nearly the same smallest average atomic volume, while the A15 structure is predicted to have the highest heat of formation and the largest average atomic volume among the structures concerned. At both compositions of Fe2 Cu and FeCu2 , the C15 structure is predicted to have the lowest heat of formation, as well as the smallest average atomic volume. At the composition of FeCu, the L10 structure, which is actu-

ally an fcc structure, has the lowest heat of formation, whereas the two bcc structures, B2 and L2a , are predicted to have a little higher heat of formation. At the composition of FeCu3 , however, the TiAl3 type structure of D022 sits in the lowest energy level, the D019 and D03 structures have a little higher energy, and the L12 and L60 structures both have even higher energies. Interestingly, one may notice that at each composition, the relatively stable structure seems to have the smallest average atomic volume. More interestingly, at the composition of FeCu, the relatively stable L10 structure has ever been obtained in experiments, i.e. by pulsed-laser deposition, one ML of Fe and one ML of Cu were alternately grown onto a Cu(1 0 0) substrate, and the obtained Fe/Cu layered structure was characterized to be in a FM state [39], which is in agreement with the present calculations. Another example is the experimental observation reported by Xu et al. that an fcc structured Fe50 Cu50 solid solution was obtained by means of ball milling and its lattice constant was measured to be ˚ [40], which was also quite close to the value of 3.63 A ˚ 3.64 A predicted for L10 (fcc) by the present calculations. From Table 2, one also sees that the Fe atoms in nearly all the possible phases possess a magnetic moment on the same magnitude of order as the HSFM state of fcc Fe discussed in section A. Referring again to Fig. 1 and Table 1, one finds that because of the involvement of Cu, the average atomic volume of the respective Fe–Cu intermetallic compound falls in the range of the HSFM state of fcc Fe, which is greater than those of the LSFM, NM, and AFM states. It is therefore believed that the relatively large magnetic moment of Fe atom in the non-equilibrium Fe–Cu alloys might be originated from the expansion of the lattice. We now take the fcc structured Fe–Cu alloys at the compositions of Fe3 Cu, FeCu and FeCu3 as examples to further examine their magnetic behaviors. Fig. 2 illustrates the variation of the average atomic volume and magnetic moment per Fe atom versus the variation of the Cu concentration. One observes that both the average atomic volume and the magnetic moment per Fe atom increase with the increasing of the Cu concentration. The observed correlation can well be explained as follows. It is known that the equilib-

Fig. 2. Variation of average atomic volume and magnetic moment per Fe atom (MM) of the fcc structured Fe–Cu alloys against the Cu concentration.

L.T. Kong, B.X. Liu / Journal of Alloys and Compounds 414 (2006) 36–41

˚ which is greater rium lattice constant of Cu is about 3.615 A, ˚ predicted for the LSFM state as well as than a = 3.463 A ˚ greater than the experimentally measured value a = 3.576 A for the ␥-Fe (fcc) precipitates in Cu at 80 K [41]. It is therefore deduced that introduction of Cu atoms into the fcc Fe lattice would cause an effective expansion of the average atomic volume, thus resulting in an enhancement of the magnetic moment of Fe. Besides, the immiscibility between Fe and Cu could also be favored to expand the atomic volume. In short, the above calculation results suggest that it is possible to stabilize the predicted HSFM state of fcc Fe in the non-equilibrium Fe–Cu intermetallic compounds, and there have indeed been some experimental studies reporting the formation of such non-equilibrium alloys in the Fe–Cu system [39,40,42,43]. 5. Summary We have shown that ab initio calculations based on the projector augmented wave method are capable of revealing the structural as well as magnetic stability of fcc Fe and predict a high-spin ferromagnetic state at large atomic volume, a coexistence of two antiferromagnetic, a non-magnetic, and a low-spin ferromagnetic state at small atomic volume. It also reveals that for the Fe atoms in the non-equilibrium Fe–Cu alloys, the high spin ferromagnetic state could retain over a broad composition range, suggesting an alternative approach to utilize the high spin ferromagnetic state of fcc Fe by alloying with some fcc structured non-magnetic elements. Acknowledgment The authors are grateful to the financial support from the National Natural Science Foundation of China, The Ministry of Science and Technology of China (G20000672), and the Administration of Tsinghua University.

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