Structure of the high-entropy alloy AlxCrFeCoNi: fcc versus bcc

Accepted Manuscript Structure of the high-entropy alloy AlxCrFeCoNi: fcc versus bcc Masako Ogura, Tetsuya Fukushima, Rudolf Zeller, Peter H. Dederichs PII:

S0925-8388(17)31539-6

DOI:

10.1016/j.jallcom.2017.04.318

Reference:

JALCOM 41722

To appear in:

Journal of Alloys and Compounds

Received Date: 24 January 2017 Revised Date:

26 April 2017

Accepted Date: 28 April 2017

Please cite this article as: M. Ogura, T. Fukushima, R. Zeller, P.H. Dederichs, Structure of the highentropy alloy AlxCrFeCoNi: fcc versus bcc, Journal of Alloys and Compounds (2017), doi: 10.1016/ j.jallcom.2017.04.318. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Structure of the high-entropy alloy AlxCrFeCoNi: fcc versus bcc

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Masako Ogura1, Tetsuya Fukushima2,3, Rudolf Zeller1 and Peter H. Dederichs1

Peter Grünberg Institute and Institute for Advanced Simulation, Forschungszentrum Jülich

and JARA, D-52425 Jülich, Germany

Institute for NanoScience Design, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka

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560-8531, Japan

Institute for Datability Science, Osaka University, Osaka 565-0871, Japan

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ABSTRACT

The effect of Al on the crystal structures of the high-entropy alloy AlxCrFeCoNi is discussed

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using first-principles electronic structure calculations. When the atomic configuration is totally random, AlxCrFeCoNi has the fcc structure. However, the total energy difference between the fcc and bcc structures decreases as the Al concentration increases. In the

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calculations Cr and Fe stabilize the bcc structure and Ni and Co work as fcc stabilizer in the alloys, as is observed in experiments. Moreover, the interactions between Al and transition

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metal elements are strongly attractive. As a result, partially disordered structures such as L12, D03 and B2, where the Al atoms are ordered and the transition metal atoms are still random, are more stable than the totally disordered phases. Especially, the energy gain by the D03 structure is large and this leads to the transition from fcc to bcc for strongly increased Al concentration.

Keywords: Transition metal alloys and compounds; Crystal structure; Electronic properties; Magnetization; Computer simulations

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1. Introduction High-entropy alloys (HEAs) [1–5] are based on a distinct concept of materials design. The conventional approach for alloys is choosing one principal element and adding small

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contributions of other ingredients so that the properties of the principal element are improved. In contrast, HEAs are designed on a multiple element basis. According to the definition by Yeh et al. [1], HEAs have five or more elements with concentrations in the 20 at.% region.

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Many types of HEAs have been investigated in the past decade and various very favorable properties have been reported, such as high hardness, good wear, oxidation and corrosion

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resistances and so on [6, 7].

Crystal structures are one of the interesting and important topics for HEAs. Most HEAs have simple solid solution structures, e.g., bcc and fcc structures. This is a result of a high entropy effect: HEAs have a very high entropy due to the disorder of all elements and this lowers the free energy for the random solid solution at high temperatures. A small amount

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of intermetallic compound phases also form in HEAs [8, 9].

In many HEAs with Al such as AlxCrFeCoNi and AlxCrFeCoNiCu, the structure changes from fcc to bcc with increasing Al concentration [1,10–12]. Along with the change of

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the structure, the hardness of these alloys increases. In the observations of microstructures in the bcc phase of these alloys, coexistence of disordered bcc (A2) and ordered bcc (B2) phases

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has been found [10,12–14]. It is pointed out by Guo et al. [15] that the stability of the bcc or fcc structure in HEAs are related to the valence electron concentration. This relation is similar to the general trend of transition metals, which is explained by the d electrons occupation [16,17]. It is also suggested that the change from fcc to bcc by insertion of Al is due to the lattice distortion. Since Al has a larger atomic size than the other principal elements, the alloys form a structure with a lower atomic-packing efficiency, say bcc [3,18]. While this sounds reasonable, the mechanism for the stability of bcc or fcc has not really been understood yet. Concerning the structures and single-phase stabilities of HEAs, computational

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ACCEPTED MANUSCRIPT attempts have been done by thermodynamic calculations [5,19], density functional theory (DFT) calculation [20–22], molecular dynamics (MD) [23], ab initio MD [14,24] and Monte Carlo simulation [25]. Some of them reproduce experimental observations well. These computational approaches are important for both understanding the physics behind these

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alloys and designing new types of HEAs.

In this paper, we concentrate on the stability of the fcc structure of the HEA CrFeCoNi and the transition to the bcc structure with increasing Al concentration from the

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viewpoint of the electronic structures in the framework of DFT. From energetic point of view the transition from an fcc structure to a bcc structure represents a first-order transition and a

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challenging problem for ab initio calculations, since the energy difference between two energy minima, which are widely separated in phase space, has to be calculated with high accuracy. The most well-known case concerns the ground state of Fe. While ab initio calculations based on the local density approximation (LDA) predict the fcc structure as ground state, calculations base on the generalized gradient approximation (GGA) yield the

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correct bcc structure for the ground state of Fe [26,27]. While the GGA functional gives excellent results for transition metals and alloys, there are also exceptions. This concerns in particular the ordered Fe3Al alloy, which has in the ground state an bcc-like D03 structure.

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This is well obtained in LDA calculations; however the GGA functional yields a wrong fcc-like L12 structure [28]. For the HEAs the accuracy problem is particular important and

small.

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highly accurate codes are required, since the differences of the structural energies are rather

2. Theoretical methods and technical details All calculations are carried out by the full-potential Korringa-Kohn-Rostoker (KKR) Green's function method. The calculations are spin-polarized and scalar relativistic. We simulate the disordered configurations by the coherent potential approximation (CPA) or by using a large supercell with 216 atoms. By means of the screened KKR technique used in KKRnano [29,30], we can perform large supercell calculations efficiently. In the supercell calculations,

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ACCEPTED MANUSCRIPT the atoms are randomly distributed on the bcc/fcc lattice and no lattice relaxation is considered. For the exchange-correlation functionals, we used different versions of GGA, however in most calculations the parametrization of Perdew, Burke and Ernzerhof (PBE) [31]. The Green's functions are expanded up to a maximum angular momentum of lmax = 4. The

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Brillouin zone integration is carried out on
 ×
 ×
 mesh points, where
 ~20/ 

  (  is the number of atoms in the unit cell). We take 36 energy points on the

complex energy contour and the electronic temperature 800 K. As a test case for the accuracy

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of our calculation, we recalculated the fcc and bcc phase of Fe, obtaining the bcc phase 13 mRy/atom lower in energy than the fcc phase, in good agreement with FLAPW calculations [26,27].

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When the whole space is divided into non-overlapping cells, the total energy  can be written as the sum of energies associated with the local contribution  of all cells n [32],  =   . 

(1)

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We call  local energy. In CPA the local energy can be written further as a weighed sum of local energies of all atom components α,

 =    , 

(2)

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where  is the concentration for the atom α on the considered site.  refers to the energy

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of the system with the local perturbation caused by an atom α embedded in the CPA medium. This decomposition of the total energy is particularly interesting, if we consider the energy difference between the random alloy with fcc and with bcc structure,    −  =   ( −  ). 

(3)

Thus the total energy difference between the two structures can be split up into the structural energy differences of the different components. While e.g. on average the fcc structure might be preferred, some components might nevertheless prefer the bcc structure. Thus we can

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ACCEPTED MANUSCRIPT single out all components as either fcc or bcc stabilizers. We consider three different magnetic states: the ferromagnetic (FM), disordered local moment (DLM) [33] and non-magnetic (NM) states. Whereas the FM calculation describes the spin alignment below the Curie temperature, the DLM calculation describes

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well the behavior above the Curie temperature. For higher temperatures the entropy also plays an important role in HEAs. However it has been shown by reference [22] that to a very large extent the entropy does not depend on the lattice structure of the alloy, so that for the problem of

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the fcc-bcc transition the entropy can be neglected.

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3. Results and discussion 3.1. FM calculations

Fig. 1(a) shows the total energy differences ∆ =  −  of AlxCrFeCoNi as a function of the molar ratio x of Al. The relation between x and the atomic Al concentration  is given by

= 4 /(1 −  ). The results of both CPA and supercell calculations for 216

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atoms are shown in Fig. 1(a). In the calculation, firstly, we evaluated the equilibrium lattice constant for each x by CPA, and then calculated the total energies with the obtained lattice constants. The equilibrium lattice constants increase as x for both bcc and fcc cases. For the

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supercell calculations for x = 1.023, which correspond to a system with 43 atoms for each transition metal and 44 Al atoms, we used the lattice constants obtained by the CPA for x = 1.

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Supercell calculations are performed for random atomic configurations for the fcc and bcc superstructures, both for x = 0 and x = 1.023. The values in Fig. 1(a) are differences between the average energy within the two configurations for bcc and fcc. The errors are evaluated from the standard deviation of these two configurations. When x = 0, the fcc structure is more stable than bcc. As x increases, the energy difference decreases. But the fcc structure is still more stable with high x, i. e. higher Al concentration. This is in disagreement with experiments, showing a transition to bcc in the range of 0.5 ≤ x ≤ 0.9 [12]. The CPA and supercell calculations show a good agreement. The energy difference between random bcc and fcc phase of about 1.8 mRy (Fig. 5

ACCEPTED MANUSCRIPT 1(a)) is rather small when compared with, for example, the 13 mRy of bulk Fe. For such a small energy difference the result could be changed by the choice of GGA functional. Fig. 1(b) shows the calculations with LDA and with PBEsol [34] and RPBE [35] GGA functionals. With PBEsol and RPBE the fcc structure is still more stable. Other GGA functionals would

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show energy differences within this range since the exchange enhancement factors for other popular GGA functionals lie around those for PBEsol and RPBE [36,37]. Results using LDA are much worse, strongly favoring the fcc phase. In the following we use the PBE functional.

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The local magnetic moments for x = 0 and x = 1 obtained in the CPA calculation are summarized in Fig. 2. The Cr moments have the opposite direction to the ones of the other

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transition metals. The local moments decrease as x increases except for the Fe moment for the fcc structure. In the supercell calculation, the Cr moments show large fluctuations. However the averaged moments of all the transition metals agree well with those obtained by CPA. For Fe, Co and Ni, the moments in the fcc phase are smaller than the ones in the bcc phase. Only the negative Cr moment is larger in fcc. It is somewhat surprising to see that in the Al-rich

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phase the moments are only slightly changed.

In order to see the contribution of each element to the total energy, we extracted the local energies. Fig. 3 shows the local energy   (α=Cr, Fe, Co, Ni or Al) for each site

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obtained by CPA. For all sites, the local energies decrease as x increases. This energy gain arises from the hybridization of the mostly occupied transition metal d-states with the higher

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lying Al-p states, leading to a strong binding as seen in Fig. 3, a charge transfer to the transition metal atoms and a reduction of the local moments (Fig. 2). For the Cr and Fe sites, the decrease of the local energy for the bcc structure is more rapid than that for fcc. Thus these components make the total energy for bcc close to that for fcc (see also Fig. 4(a)). Especially, at the Cr site, the local energy for bcc is always lower than that for fcc. At the Fe site, the local energy for bcc becomes lower than that for fcc as x increases. The trend of the local energies are more clearly seen in Fig. 4(a), where the local   energy differences ∆  =  −  for x = 0 and x = 1 are shown. This trend does not

change much even if we assume different concentrations for transition metals. In particular

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ACCEPTED MANUSCRIPT we see that for increasing Al concentration Cr and Fe favor the bcc phase, while the strong favoring of fcc by Co is reduced. To see the effect of the stabilizers for fcc or bcc in a realistic calculation we show in Fig. 4(b) the total energy difference between fcc and bcc by varying the concentration of the components α. By increasing the concentration of Fe and Cr the bcc structure

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becomes more stable. It has been reported that Ni works as a fcc stabilizer and Cr is a bcc stabilizer in the experiments [38]. This trend is reproduced in our calculations. The energy differences of the different transition metals exhibit a similar behavior as is found in the bulk

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of these metals, i.e. Co and Ni prefer compact fcc structures (or hcp in case of Co), while Fe and Cr prefer open bcc-like structures. For these reason we believe that an exchange of Cr by

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Mn or V would lead to a similar behavior.

The local energy reduction at the transition metal sites by Al is also seen in the results obtained by the supercell calculation. Fig. 5 shows the local energy fluctuation #  =   −

$$$$   for x = 1.023 as a function of the number of Al atoms in the nearest-neighbor distance.  is the average local energy of the element α. We can see that transition metals surrounded $$$$ 

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by more Al atoms have lower local energies. The range of the fluctuations is rather large compared to that in the system without Al.

The local energy decreases more in the bcc lattice by the same number of Al

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neighbors than the fcc lattice, as is seen in Fig. 5. This might be because of the closer distance of the nearest-neighbor site in the bcc structure. However, in the fcc structure, the transition

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metal atoms have, for the same Al concentration, more Al neighbors than bcc. Thus the stability of bcc or fcc structure is determined by the competition of these two effects, closer distance or more Al neighbors. This makes it difficult to explain the difference of total energy changes in these structures by insertion of Al.

3.2. DLM-CPA calculations Fig. 6(a) shows the total energy difference between the DLM/NM and FM states. From theoretical point of view the Curie temperature &' can be roughly evaluated from the energy differences between the FM and DLM states [39], 7

ACCEPTED MANUSCRIPT 2 () &* ≅ (-./ − 0/ ). 3

(4)

The evaluated Curie temperature is several hundred K and it is the same order with experiments and the other theoretical work [40,41]. The fcc structure shows lower Curie

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temperature than bcc and the Curie temperature decreases as x. These behaviors agree with experimental observation [40].

The total energy differences between the bcc and fcc structures are given in Fig.

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6(b) for the FM structures, the DLM structures as well as for the NM structures. The energies of all three methods shift with increasing Al-content towards the bcc structure, but stay in fcc

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even for large Al concentrations. The differences between the FM, DLM and NM states are due to magnetism. The occurrence of magnetism basically counteracts the simple picture of the p-d hybridization, since the magnetism needs empty 3d states above the Fermi energy EF, while the p-d hybridization gains energy by pushing these states below EF. This and the fact that the magnetism is different in the fcc and bcc structures complicate the interpretation of

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the data.

The local moments obtained by the DLM method for the bcc and fcc structures are shown in Fig. 7 and should be compared with the moments in Fig. 2 obtained by FM-CPA.

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Firstly one sees that the moments in the DLM calculations are somewhat smaller than the moments in the FM calculations. However the moments for Cr and Ni completely vanish for

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both structures and moreover the moment of Co in the fcc structure is very small. This effect is typical for DLM and very similar to the moments of magnetic impurities in e.g. noble metals. A moment can only exist, if the local density of states of the impurity at the Fermi energy is larger than the reciprocal of the exchange integral of the considered atom. The calculation shows, that this is not the case for Cr and Ni and the local DOS at EF of Co in fcc is only slightly larger than the critical value, so that the moment is very small in the fcc structure. The reduction of the moment in the DLM calculation as compared to the FM calculation basically explains why the energy differences ∆ =  −  in Fig. 6(b) are lower than the FM values and slightly move in the direction of the NM values.

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3.3. Partially disordered structures The local energy reduction shown in Fig. 3 due to surrounding Al indicates that it is energetically more favorable when the Al atoms are distributed such that more transition

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metal atoms are next to Al. Therefore we performed CPA calculations of partially disordered systems. We considered B2, D03, and two types L12 structures. The B2 and D03 structures are bcc like and the L12 structure is fcc like. These structures are illustrated in Fig. 8.

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The total energies of these partially disordered systems relative to the energy of the totally disordered bcc structure obtained by the FM-CPA are shown in Fig. 9. These partial

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disordered structures have lower energy than the completely disordered structures. Especially, the B2, D03 and L12-1 structures show a large energy gain compared to fully disordered bcc and fcc. Although the L12-1 structure is more stable than the bcc-like structures for small x-values, D03 becomes more stable at x ≥ 1.

We also considered the coexistences of partially disordered and totally disordered

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structures. We call the coexistence of B2 and the random bcc structures B2+bcc. The B2 phase is Al(CrFeCoNi), i.e., one site of the B2 structure is occupied by Al and the other site is randomly occupied by the transition metals. All Al atoms are in the B2 phase and the rest of

given by

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the transition metals make a random bcc phase. The total energy of this new phase B2+bcc is

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)12 = 2 )13(*405* 67) + (1 − 2 )3*405* 67 ,

(5)

with the atomic Al concentration cAl. Similarly we considered the coexistence of D03-type Al(CrFeCoNi)3 and the random bcc CrFeCoNi (D03+bcc) whose total energy is given by -9 2 = 4 -9 3(*405* 67) + (1 − 4 )3*405* 67 .

(6)

Another system concerns the coexistence of L12-type Al(CrFeCoNi)3 and random fcc CrFeCoNi (L12+fcc). The total energy for this case is written as .:; 2 = 4 .:;3(*405* 67) + (1 − 4 )3*405* 67 .

(7)

The total energies for these three cases are also shown in Fig. 9. These three systems B2+bcc, D03+bcc and L12+fcc have lower energies than B2, D03 and L12-1, respectively, and much 9

ACCEPTED MANUSCRIPT lower energies than the random bcc and fcc phases. These results are consistent with experiments [12–14] and MD simulations [14,24] for the bcc phase, where some segregations and spinodal decompositions of two or more phases are observed. For fcc, phases with the L12 structure are also observed by experiments [42]. The total energies for D03+bcc and

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L12+fcc also cross at x = 1.

Note that all these energies are calculated by the FM-CPA method. However calculations by the more relevant DLM-CPA give very similar results. Analogous to the shift

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of the energy curves in Fig. 6(b) all energy curves of the fcc-like structures in Fig. 9 are shifted down-ward in the DLM calculation. In particular the crossing between the L12+fcc

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and D03+bcc energy curves occur in the DLM-calculations at x ~ 1.33.

Thus, also in our calculation, the transition from fcc-based structure to bcc-based structure takes place. However the calculated transition concentration is larger than experimentally observed 0.5 ≤ x ≤ 0.9. One possible reason is an error of the GGA-PBE functional. Also the effect of lattice relaxations, which are neglected in this calculation, is

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important. Despite of these uncertainties, we believe that the important trends found in our calculations are generally valid. In particular this concerns the identification of bcc and fcc

4. Summary

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stabilizers and the strong stability of the above partially disordered phases.

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We performed first-principles calculations for the AlxCrFeCoNi HEA and discussed the effect of Al on its structure. We considered both the structural disorder as well as the spin disorder. Clearly Al addition leads to a reduction of the bcc energies, while the fcc structure of CrFeCoNi has lower total energy than the bcc one. The local energies show that Cr and Fe stabilize the bcc structure and Ni and Co act as fcc stabilizers. Al strongly reduces the local energy of the transition metal atoms by p-d hybridization. In addition to the random structures we also consider partially disordered structures such as D03, B2 and L12, which are more stable than totally disordered structures. The energy gain by making the D03 structure is especially large and the fcc-bcc transition takes place with increasing the Al concentration.

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ACCEPTED MANUSCRIPT Thus for larger Al concentrations, the bcc-like D03 phase is stabilized, leading to a segregation into the partially disordered Al-rich D03 phase and a totally disordered bcc phase of CrFeCoNi.

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Acknowledgments

The authors gratefully acknowledge the computing time granted by the JARA-HPC Vergabegremium and provided on the JARA-HPC Partition part of the supercomputer

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JUQUEEN at Forschungszentrum Jülich. T. F. thanks the supports from “Professional

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development Consortium for Computational Materials Scientists (PCoMS)”

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[35] B. Hammer, L. B. Hansen, J. K. Nørskov, Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals, Phys. Rev. B 59 (1999) 7413–7421.

[36] P. Haas, F. Tran, P. Blaha, Calculation of the lattice constant of solids with semilocal

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functionals, Phys. Rev. B 79 (2009) 085104.

[37] K. Yang, J. Zheng, Y. Zhao, D. G. Truhlar, Tests of the RPBE, revPBE, τ-HCTHhyb, ωB97X-D, and MOHLYP density functional approximations and 29 others against

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representative databases for diverse bond energies and barrier heights in catalysis, J. Chem. Phys. 132 (2010) 164117.

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[38] B. Ren, Z. Liu, D. Li, L. Shi, B. Cai, M. Wang, Effect of elemental interaction on microstructure of CuCrFeNiMn high entropy alloy system, J. Alloys Compd. 493 (2010) 148– 153.

[39] P. H. Dederichs, K. Sato, H. Katayama-Yoshida, Dilute magnetic semiconductors, Phase Transition 78 (2005) 851–867. [40] Y.-F. Kao, S.-K. Chen, T-J. Chen, P.-C. Chu, J.-W. Yeh, S.-J. Lin, Electrical, magnetic, and Hall properties of AlxCoCrFeNi high-entropy alloys, J. Alloys Compd. 509 (2011) 1607– 1614. [41] F. Körmann, D. Ma, D. D. Belyea, M. S. Lucas, C. W. Miller, B. Grabowski, M. H. F.

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ACCEPTED MANUSCRIPT Sluiter, “Treasure maps” for magnetic high-entropy-alloys from theory and experiment, Appl. Phys. Lett. 107 (2015) 142404. [42] T.-T. Shun, Y.-C. Du, Microstructure and tensile behaviors of FCC Al0.3CoCrFeNi high

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SC

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entropy alloy, J. Alloys Compd. 479 (2009) 157–160.

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0

5

10

at.% of Al 15 20

25

30

0 (a)

-2 -3 CPA supercell

-4 -5 0

SC

(b)

-4 -6

PBE RPBE PBEsol LDA

M AN U

E (mRy/atom)

-2

RI PT

E (mRy/atom)

-1

-8 -10 0.0

0.5

1.0

1.5

x

Fig. 1. (a) Total energy differences ∆ =  −  of AlxCrFeCoNi calculated by CPA and

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the supercell method. Here the GGA-PBE functional is used. (b) Total energy differences

bcc

2

fcc

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Local magnetic moment (µ B)

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calculated by CPA for LDA and three types of GGA functionals.

1

0 x=0 x=1

-1 Cr

Fe

Co

Ni

Cr

Fe

Co

Ni

Fig. 2. Local magnetic moments of AlxCrFeCoNi obtained by CPA.

16

ACCEPTED MANUSCRIPT 0 bcc fcc

(a) Cr -20

Local energy (mRy)

-40 -60

(b) Fe

-80

RI PT

0 (c) Co

-20

(d) Ni

-40

-80 0.0

0.4

0.8

1.2

1.6 0.0 x

0.4

SC

-60

0.8

1.2

1.6

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Fig. 3. Local energies relative to the value for bcc and x = 0 in AlxCrFeCoNi calculated by CPA. The closed and open circles are for the bcc and fcc structures, respectively.

0

10

x=0 x=1

30

(b)

E (mRy)

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α

E (mRy)

at.% of Al 20

2

5 0

-5

-10

10

0 -2 (CrFe)1.5(CoNi)0.5Alx (CrFe)1.4(CoNi)0.6Alx CrFeCoNiAl x

-4

(a)

-15

Fe

EP

Cr

Co

0.0

Ni Al

0.5

1.0 x

1.5

  Fig. 4. (a) Local energy differences ∆  =  −  of AlxCrFeCoNi. (b) Total energy

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differences ∆ =  −  of Alx(CrFe)1+y(CoNi)1-y.

17

ACCEPTED MANUSCRIPT 40 bcc fcc

(a) Cr

0 -20 (b) Fe -40 40

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Local energy fluctuation (mRy)

20

(c) Co

(d) Ni

20 0

SC

-20 -40 1

2

3 4 5 0 1 2 3 Number of nearest-neighbor Al atoms

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0

4

5

Fig. 5. Local energy fluctuations in Al1.023CrFeCoNi calculated by the supercell calculation.

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The solid lines indicate the average values.

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0

5

10

at.% of Al 15 20

25

30

12 (a)

8

1000 800 600

4

400

2

200

0 0

0

RI PT

6

(b)

SC

-2 -4

E (mRy/atom)

1200

T (K)

E (mRy/atom)

10

NM bcc NM fcc DLM bcc DLM fcc

-8 -10

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-6

FM NM DLM

-12 -14 0.0

0.5

1.0

1.5

x

6/ 0/ 6/ 0/ Fig. 6. (a) Total energy differences ∆ =  −  (closed circles),  −  (open

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-./ 0/ -./ 0/ circles),  −  (closed squares) and  −  (open squares) of AlxCrFeCoNi

calculated by CPA. (b) Total energy differences ∆ =  −  for FM, NM and DLM

Local magnetic moment (µ B)

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states.

bcc

2

fcc

1

0 x=0 x=1 -1 Cr

Fe

Co

Ni

Cr

Fe

Co

Ni

Fig. 7. Local magnetic moments of AlxCrFeCoNi for the DLM state.

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D03 (x < 4/3)

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D03 (x > 4/3)

CrFeCoNi L12 -1 (x < 4/3)

AlCrFeCoNi

SC

B2

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Al

L12 -2

L12 -1 (x > 4/3)

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Fig. 8. Crystal structures of partially disordered AlxCrFeCoNi considered in this study.

0

5

10

15

at.% of Al 20

25

30

0

-4

-6

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E (mRy/atom)

EP

-2

-8

-10 0.0

fcc L12-1 L12-2 L12+fcc

B2 B2+bcc D03 D03+bcc 0.5

1.0

1.5

x

Fig. 9. Total energy differences ∆ =  −  of FM totally/partially disordered AlxCrFeCoNi.

20

ACCEPTED MANUSCRIPT

Stability of bcc/fcc is discussed from the local contribution of total energy.

l

Al reduces the local energy of the transition metal atoms by p-d hybridization.

l

Partially disordered structures are stable and the fcc-bcc transition takes place.

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EP

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M AN U

SC

RI PT

l