Structural, electronic and magnetic properties of Ni2XAl (X= V, Cr, Mn, Fe, and Co) Heusler alloys: An ab initio study

Structural, electronic and magnetic properties of Ni2XAl (X= V, Cr, Mn, Fe, and Co) Heusler alloys: An ab initio study

Journal Pre-proof Structural, electronic and magnetic properties of Ni2XAl (X= V, Cr, Mn, Fe, and Co) Heusler alloys: An ab initio study Yin-Kuo Wang,...

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Journal Pre-proof Structural, electronic and magnetic properties of Ni2XAl (X= V, Cr, Mn, Fe, and Co) Heusler alloys: An ab initio study Yin-Kuo Wang, Jen-Chuan Tung PII:

S2666-0326(19)30008-0

DOI:

https://doi.org/10.1016/j.physo.2019.100008

Reference:

PHYSO 100008

To appear in:

Physics Open

Received Date: 11 November 2019 Accepted Date: 20 November 2019

Please cite this article as: Y.-K. Wang, J.-C. Tung, Structural, electronic and magnetic properties of Ni2XAl (X= V, Cr, Mn, Fe, and Co) Heusler alloys: An ab initio study, Physics Open, https:// doi.org/10.1016/j.physo.2019.100008. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Published by Elsevier B.V.

Structural, electronic and magnetic properties of Ni 2 XAl (X= V, Cr, Mn, Fe, and Co) Heusler alloys: An ab initio study Yin-Kuo Wanga, Jen-Chuan Tungb a

Graduate Institute of Mass Communication, National Taiwan Normal University, Taipei ,106 Taiwan

b

Center for General Education, China Medical University, Taichung 40402, Taiwan

Email address: [email protected] (Jen-Chuan Tung) Abstract Nickel-based Ni 2 XAl (X= V, Cr, Mn, Fe, and Co) Heusler alloys for both L2 1 and L1 0 crystal structures are studied by using the density functional theory. The magnetization energy and formation energy of Ni 2 XAl Heusler alloys are calculated. We found that in the ideal L2 1 structure, ferromagnetic (FM) state is more stable than nonmagnetic (NM) and antiferromagnetic (AF) states, except Ni 2 VAl. The ground state for Ni 2 VAl is the nonmagnetic state. The so-called Bain paths method is applied to study the stability of Ni 2 XAl alloys under tetragonal distortion. We found that Ni 2 FeAl and Ni 2 CoAl are possible ferromagnetic shape materials (FSMs). We also found that Ni 2 CrAl and Ni 2 CoAl are energetically favorable in the antiferromagnetic state in the L1 0 structure. To study the stability between L2 1 and L1 0 crystal structure, we calculate the cohesive energy for comparison. We also calculate the elastic constants, bulk modulus, shear modulus, and universal elastic anisotropy to study the mechanical stability in L2 1 and L1 0 crystal structure. The spin polarization for the L2 1 structure is calculated whilst the largest spin polarization is smaller than 80 % .

Keywords: density functional theory, Ni-based Heusler alloy, elastic constants

1. Introduction Since the discovery of Ni 2 MnGa Heusler alloy[1], many Heusler alloys become promising intermetallic materials for both theoretical and experimental interests due to their potential applications in not only spintronic devices but also fundamental research field. For example, Mn 2 NiIn[2], Ni 2 MnAl[3, 4, 5, 6], Ni 2 MnGa[7] and Ni 2 FeGa[8] are ferromagnetic shape memory alloys. The ferromagnetic shape memory alloys are those materials who can change shape to its original one by applying an external magnetic field. These FSMAs will undergo a phase transition, called martensitic transformations, from cubic to orthorhombic structure when cooling or heating.[8] The chemical formula for the so-called full-Heusler alloy is X 2 YZ, where X and Y are transition metals and Z is usually the main group element. The physical properties of Heusler alloys such as superconductivity[9], Hall and anomalous Hall effect[10], half-metallic ferromagnetism[10] are studied experimentally and theoretically in the past decades. The ideal crystal structures of the Heusler alloys are cubic L2 1 structure. Ni-based Heusler alloys, such as Ni 2 MnGa[7] and Ni 2 MnAl[3], have recently received much attention in ferromagnetic shape memory applications. These Ni-based Heusler alloys will undergo a structural phase transition from the high-temperature cubic phase to low temperature tetragonal distorted one. For example, Ni 2 FeGa alloy was found can change phase from cubic L2 1 to the orthorhombic L1 0 structure at 142K, i.e., martensitic-austenite transition. The physical properties of Heusler alloys can be easily tuned by composition. The key ingredients in these Heusler alloys are a high spin polarization, large magnetocrystalline anisotropy energy (MAE) or a large magnetic shape memory effect.

New Heusler compounds are usually investigated theoretically by using the density functional theory. These Heusler alloys are theoretically suggested to be stable especially in the ferromagnetic state. However, the stability of these compounds is mostly considered merely the formation energy, and/or cohesive energy. Another important criterion for stability is mechanical stability, i. e., the elastic constants, whereas it provides important information when concerning the strength of materials. In this paper, we perform a systematic study in the structural, electronic, elastic, and magnetic properties of Ni 2 XAl (X= V, Cr, Mn, Fe, and Co) Heusler alloys for both L2 1 and L1 0 crystal structures and nonmagnetic and magnetic states. We firstly describe the computational method in section 2. The structural and magnetic properties are discussed in section 3.1. The density of states (D.O.S.) is plotted and discussed in section 3.2. The last part of section 3 is the mechanical stability of L2 1 and L1 0 structures. Figure 1: (color online) Crystal structure of the (a) ideal L2 1 structure and (b) tetragonal L1 0 structure of the Ni 2 XAl Heusler alloys. Table 1: Calculated Ni 2 XAl Heusler alloys lattice constant a (Å), spin magnetic moments mX ,

mY , and mZ ( μB ), total magnetic moments per formula unit mt ( μB ), formation energy E f (eV),

(

magnetization energy ΔE = E

FM ( AF )

)

− E NM (eV), and cohesive energy Ecoh (eV) in the L2 1

crystal structure. L2 1 crystal structure a

mNi

mY

mt

mAl

Ef

Ecoh

eV

μB

Å

ΔE

Ni 2 VAl NM

5.793

-1.534

0

20.183

-0.581

0

17.845

-1.063

-0.482

18.327

5.8031a, 5.800b Ni 2 CrA l NM

5.735 5.74

FM

AF

c

5.792

0.314

2.556

-0.063

3.212

5.800c,5.821d, 5.78e

0.26c

2.39c

-0.03c

3.503d

5.803

0.023

2.721

0.044

0.000

-1.041

-0.460

18.305

5.700









-0.579

0

17.543

-0.381c

Ni 2 Mn Al NM

FM

5.789

0.362

3.364

-0.073

4.046

5.789f, 5.81g

0.36g

3.35g

-0.038g

4.059f, 4.07g

5.791

0.002

3.358

0.040

0.000

-1.681

-1.102

18.645

NM

5.677









-0.465

0

18.856

FM

5.743

-1.311

-0.846

19.702

AF

-1.763

-1.184

18.727 17.82g

Ni 2 FeA l 0.252

2.786

-0.051

3.217

5.744 , 5.75 , 5.70h

0.24

g

2.57

-0.024

3.385 ,2. 99g, 3.39h

5.744

0.008

2.764

0.007

0.000

-1.232

-0.767

19.623

NM

5.665









-0.939

0

19.740

FM

5.689

0.113

1.460

-0.031

1.611

-1.048

-0.109

19.849

-1.028

-0.089

19.829

d

AF

g

g

g

18.90g

d

Ni 2 Co Al

5.688 , 5.604 d

AF a

1.791 ,1. 78i

i

d

5.687

0.011

1.388

0.012

0.000

Ref. [15]; Ref. [9]; Ref. [16]; Ref. [3]; Ref. [17]; Ref. [6]; Ref. [7]; Ref. [18]; Ref. [19]. b

c

d

e

f

g

h

i

2. Structure and computational method The ideal Ni 2 XZ (X= V, Cr, Mn, Fe, and Co) full-Heusler alloys are cubic L2 1 structure with point group 225 ( Fm 3 m symmetric), see Fig 1.(a). It consists of four interpenetrating fcc lattices with two Ni atoms, Y atom, and Z atom are placed on the (1/2, 1/2, 1/2), (0, 0, 0), (1/4, 1/4, 1/4), and (3/4, 3/4, 3/4), respectively. These two Ni atoms are equivalent in the primitive cell as shown in Fig. 1(a). The cubic L2 1 structure can also be described by a L1 0 type tetragonal phase (No. 139, I 4 / mmm ) with a = b , c = 2a , where Ni, Y and Z atoms are placed in (1/2, 0, 1/4), (1/2, 1/2, 0) and (0, 0, 0), respectively. The structural, electronic and magnetic properties of Ni 2 XAl Heusler alloys are calculated under the frameworks of density functional theory as implemented in the Vienna ab initio simulation package (VASP)[11, 12] with the generalized gradient approximation (GGA)[13, 14] for the exchangecorrelation functional. The Γ -centered Monkhorst-Pack scheme with a k-mesh of 12× 12× 12 is used for the Brillouin zone integration. The self-consistent total energy criterion is set to be 1.0× 10 −6 eV. To find the equilibrium lattice constant a of the cubic L2 1 structure, we calculated the total energy as a function of the lattice constant. The theoretical lattice constant for the nonmagnetic, ferromagnetic and antiferromagnetic states in the L2 1 structure is obtained from the minimum of the total energy. In the L1 0 tetragonal distorted phase, the total energy can not be described merely as a function of the lattice constant because of the structure parameters have two degrees of freedom, i. e. lattice constant a and c / a ratio. Previous theoretical studies in Ni 2 MnGa, Mn 2 NiGa, Ni 2 CrAl, Ni 2 FeAl, and Ni 2

CoAl[3] put more attention on the ferromagnetic shape memories effect, therefore, the so-called Bain paths method is applied to investigate the physical properties between two intermediate tetragonal phases, especially from L2 1 to L1 0 phase transition. In this method, it is assumed that the volume is kept in constant under varies deformation of the cubic phase. Nonetheless, the volume that used dominates the calculation results[6]. Beyond the Bain paths method, we also calculate the total energy by using the structure as demonstrated in Fig. 1(b). Firstly, the c / a ratio is chosen and fixed, we obtain the equilibrium lattice constant in the minimum of the calculated total energy. Secondly, the c / a ratio is scanned from 1.00 to 1.70. Finally, we obtain the total energy as a function of c / a ratio. In this approach, the c / a ratio is artificially fixed, therefore, the volume and the obtained physical properties such as spin magnetic moments can differ significantly with the corresponding one calculated from the Bain path method. The results for the L1 0 phase are presented in Table 2. To study mechanical stability, the elastic tensor Cij is calculated using a stress-strain methodology also implemented in VASP[11, 12].

3. Results and discussion 3.1. Structure and magnetic properties Listed in Table 1 is the calculated lattice constant, a (Å), Ni, X, Al and total spin magnetic moments ( μB ), formation energy E f , magnetization energy ΔE , and cohesive energy Ecoh (eV) for the Ni 2 XAl Heusler alloys in the L2 1 structure. It is seen that the calculated lattice constant a for both the ferromagnetic and antiferromagnetic states is larger than the corresponding one in the nonmagnetic state due to the spin-polarized effect, except Ni 2 VAl, where no stable FM and AF states are found. However, the differences in lattice constant between magnetic and nonmagnetic state are very small, being less than ∼ 1 % . The magnetization energy is defined as the total energy difference between magnetic and nonmagnetic states. We found that the ground state for the Ni 2 XAl Heusler alloys is all ferromagnetic state, except Ni 2 VAl. Nonetheless, the differences in magnetization energy between FM and AF states are very small. Ni 2 MnAl has the largest FM and AF magnetization energy of 1.184 and -1.102 (eV), respectively. Recently, both ferromagnetic and antiferromagnetic ground states in the Ni 2 MnAl Heusler alloy are reported in experiments. Table 1 also shows the calculated formation energy and cohesive energy from the formula

E f = Etot − 2 ENifcc − EYbulk − E Alfcc ,

(1)

Ecoh = ∑Eatom − Etot , (2) where ENifcc , EYbulk and E Alfcc is the total energy per atom for Ni, Y and Al, respectively, and Eatom is the isolated energy of the atom. The formation energy for all calculated Heusler compounds in all the nonmagnetic and magnetic states are negative and hence are thermally stable from the enthalpy point of view. Ni 2 MnAl in the ferromagnetic state has the lowest formation energy of -1.763 eV. Furthermore, the cohesive energy is the energy required to break alloys into isolated atoms. If the cohesive energy is low, meaning that the atoms in the alloys are more mobile than those in high cohesive energy. In other words, the alloys of high cohesive energy are more stable. Ni 2 VAl has the largest cohesive energy in this study. Table 2: Calculated Ni 2 XAl Heusler compound lattice constant a (Å), c / a ratio, spin magnetic moments for mX , mY , and mZ ( μB ), total magnetic moments mt ( μB /f.u.), formation energy, magnetization energy and cohesive energy (eV) in the L1 0 crystal structure.

L1 0 crystal structure a

c/a

mNi

mY

mAl

mt

Ef

ΔE

Ecoh

eV

Å



4.1 0

1.4 1









-1.534

0

20.183

NM

4.0 4

1.4 3









-0.582

0

17.846

FM

4.1 0

1.4 1

0.315

2.560

0.063

3.220

-1.064

-0.482

18.328

AF

3.8 6

1.6 9

0.052

2.762

0.035

0.000

-1.113

-0.532

18.378

NM

3.6 3

1.9 2









-0.732

0

17.696

FM

4.1 1

1.4 1

0.371

3.372

0.071

4.073

-1.764

-1.032

18.728

AF

4.5 8

1.4 4

0.015

3.344

0.042

0.000

-1.681

-0.950

18.645

NM

4.1 9

1.2 4









-0.549

0

18.940

FM

4.0 7

1.4 1

0.254

2.789

0.051

3.224

-1.311

-0.817

19.702

4.9 5

1.3 5a

0.39a

2.38a

-0.031a

3.08a

5.2 1

1.3 4b

-1.291

-0.797

19.681

μB

Ni 2 VAl NM Ni 2 CrA l

Ni 2 Mn Al

Ni 2 FeA l

3.29b

1.3 5c AF

4.2 1

1.2 7

0.040

2.750

0.020

0.000

Ni 2 Co Al NM

4.1 2

1.3 0









-0.953

0

19.754

FM

4.1 9

1.2 5

0.210

1.448

0.039

1.777

-1.090

-0.137

19.891

5.1 1

1.3 8b

-1.095

-0.143

19.896

2.00b

1.3 8c AF a

4.1 9

1.2 5

0.108

1.501

0.016

0.000

Ref. [7]; b Ref. [3]; c Ref. [22].

Calculated ground state lattice constant is 5.793(NM), 5.792(FM), 5.789(FM), 5.743(FM), and 5.689(FM) Å for Ni 2 VAl, Ni 2 CrAl, Ni 2 MnAl, Ni 2 FeAl, and Ni 2 CoAl, respectively. The differences in lattice constant can roughly be referred to the differences in the volumes of X atoms. Our calculated lattice constants in the ferromagnetic L2 1 structure are in very good agreement with previous theoretical calculations. For example, in Ni 2 VAl Heulser alloy, our calculated lattice constant is 5.793 Å, which is in very good agreement with 5.803[15] Å and 5.800[9] Å. In the ferromagnetic Ni 2 CrAl, our calculated lattice constant is 5.792 Å, which is also in very good agreement with 5.821[3] Å and 5.78[17] Å. For the Ni 2 MnAl, our calculated lattice constant is 5.789 Å, which is also in very good agreement with 5.789[6] Å, and 5.81[7] Å. Further, for the Ni 2 FeAl, our calculated lattice constant is 5.743 Å, which is in very good agreement with 5.744[3] Å , 5.70[18] Å, and 5.75[7] Å. For the Ni 2 CoAl, our calculated lattice constant is 5.689 Å, which is almost identical to Ref. [1] and is in good agreement with 5.604[19] Å. Calculated total spin magnetic moments for the ferromagnetic Ni 2 CrAl, Ni 2 MnAl, Ni 2 FeAl and Ni 2

CoAl in the L2 1 structure are 3.212, 4.046, 3.127 and 1.611 ( μB /f.u.), respectively. The mX is the

largest among Ni 2 XAl indicating that the ferromagnetism is totally from the X site. Besides, the mAl always gives raise a small and negative contribution to the spin magnetic moment. The spin magnetic moments for the Ni atom in both FM and AF states are in the range of 0.001 to 0.36 ( μB ), this is in good agreement with previous experimental[20, 21] and theoretical[8] results in the Ni-based Heusler alloys. Very interestingly, the spin magnetic moments of Ni atom in the FM state is one order of magnitude larger than the corresponding one in the AF state. To gain further insight into the martensitic-austenite transformation, plotted in Fig. 2 is the total energies as a function of c / a ratio for the Ni 2 XAl Heusler alloys. Total energies in the cubic L2 1 structure is set to be zero as references. It is seen in Fig. 2(a), Ni 2 FeAl and Ni 2 CoAl have two energy minimums at c / a = 1.00, 0.88 and 1.34, 1.38, respectively, whilst Ni 2 CrAl and Ni 2 MnAl have only one minimum at c / a = 1.00. Suggesting that Ni 2 FeAl and Ni 2 CoAl are possible ferromagnetic shape memory alloys. These results are also reported in the previous theoretical calculation[3]. Nonetheless, a previous theoretical study in Ni 2 MnAl[6] found that the results of the Bain path in different volume ratios may be quite different. We also calculate the total energies as a

function of c / a ratio in the antiferromagnetic state of Ni 2 XAl Heusler alloys for comparison and the results are plotted in Fig. 2(b). We found that Ni 2 CoAl has two minimums at c / a = 0.93 and 1.34, whilst Ni 2 CrAl, Ni 2 MnAl, and Ni 2 FeAl have one minimum at c / a = 1.00, 1.00, and 1.15. Ni CoAl Heusler alloy is the only one that cubic L2 1 structure is metastable for both ferromagnetic and antiferromagnetic states. 2

Listed in Table 2 is the calculated lattice constant, Ni, Y, Al and total spin magnetic moments, formation energy, magnetization energy, and cohesive energy for the Ni 2 XAl Heusler alloys in the tetragonal distorted L1 0 structure. We first notice that the stable crystal structure for the Ni 2 VAl, Ni 2

MnAl, and Ni 2 FeAl are cubic L2 1 phase, where the calculated ground state c / a ratio is 1.41 and

the calculated lattice constant are almost identical to the corresponding one in the L2 1 structure, see Table 1 and Table 2. Ni 2 CrAl and Ni 2 CoAl are stabled in the antiferromagnetic state and the c / a ratio is 1.69 and 1.25, respectively. We found that in the L1 0 tetragonal phase, almost all the ferromagnetic Ni 2 XAl are stabled in the L2 1 structure, except Ni 2 CoAl, whilst all antiferromagnetic state are stabled in the L1 0 phase. The spin magnetic moment of Ni atom is also in the range of 0.01 to 0.37 ( μB ) and Y atom gives the largest contribution to the total spin magnetic moments for both ferromagnetic and antiferromagnetic states. Furthermore, calculated spin magnetic moment of Ni atom in the antiferromagnetic state is around 0.1 μB or less which is one order of magnitude smaller than the corresponding one in the ferromagnetic state. The spin magnetic moment of the Y atom is about the same for both FM an AF state and the spin magnetic moment of Al in the AF state is about half of that in the ferromagnetic state. Interestingly, ferromagnetic Ni 2 CrAl also has the smallest cohesive energy. Figure 2: (color online) Total energies and spin magnetic moments as a function of the c / a ratio for the Ni 2 XAl in the (a) ferromagnetic (left panel), and (b) antiferromagnetic (right panel) state. The spin magnetic moments in (d) is mX .

3.2. Density of states Figure 3: (color online) The density of states of the L2 1 structure (left panel), and L1 0 structure (right panel) of the Ni 2 XAl Heusler alloys in the ferromagnetic state. The Fermi level is shifted to zero. Plotted in Fig. 3 is the densities of states of the Ni 2 XAl alloys in the ferromagnetic states. It is seen that all Ni 2 XAl Heusler alloys are metals since both spin majority and spin monority states are non zero at the Fermi level. Comparing the spin up densities of states of Ni 2 CrAl(Fig 3(a)), and Ni 2 MnAl(Fig 3(c)), due to an additional valence electron, the change in densities of state can be regarded as only the spin majority density curves shift to left whilst the spin minority density of state curve remains the same. Also, Ni and X atoms give raise main contributions to the density of states at the Fermi level and its orbital components are d-orbitals. There is a gap in the spin-down density of states near the Fermi level, indicating that these Heusler alloys are possibly half metals if the Fermi level is shifted to within that gap. Clearly, the differences between Fig. 3(c) and Fig. 3(e) can roughly be regarded as spin-down density of state curve moves to left, due to Mn atom has 5 d-electrons. This can be explained by Hund’s rule. In the ferromagnetic materials, the spin polarization is an interesting property for both experimental and theoretical studies. A high spin-polarized material can be used for providing a pure spin current

source and hence is a key ingredient for the spintronic device. Typically, the spin polarization is described in terms of the spin-decomposed densities of states at Fermi level from

N↑ − N↓ P= ↑ , N + N↓

(3)

where N ↑ ( N ↓ ) represents the numbers of spin up(down) densities of states at Fermi level. Many Heusler alloys are theoretically predicted to have 100 % spin polarization. This theoretical definition of spin polarization may differ significantly from different experimental results because of the overestimate of the contribution in the spin current of d-electrons. The spin polarization defined by Eq. (3) varies from -1.0 to 1.0. Our calculated spin polarization for the Ni 2 CrAl, Ni 2 MnAl, Ni 2 FeAl and Ni 2 CoAl in the L2 1 structure is -1.24 % , -29.14 % , -50.76 % , and -78.89 % , respectively and this spin polarization becomes smaller in the L1 0 phase. Table 3: Calculated Ni 2 XAl Heusler alloys elastic constant C11 , C12 , C44 , average bulk modulus B = (BV+ BR)/2, average shear modulus G = (GV+ GR)/2 (in GPa), universal elastic anisotropy AU and B/G ratio in the L2 1 crystal structure. L2 1 crystal structure C 11

C 12

C 44

B

G

GPa

AU

B/G





Ni 2 VAl NM

205.3

168.7

109.0

180.87

105.82

18.98

1.71

181

a

Ni 2 CrAl NM

189.7

185.9

115.6

187.18

42.58

83.28

4.40

FM

172.7

153.5

120.0

159.87

75.49

25.23

2.12

153.16b 189.3c

164.6c

109.4c

172.8c 150.0d

AF

164.9

148.1

118.1

156.19

18.63

29.59

8.53

NM

162.0

200.6

119.0

187.70

-50.08

-1.52



FM

171.6

148.3

117.2

156.03

82.84

22.65

1.88

174.84e 151.70e 115.43e 159.41e

73.89e

Ni 2 MnAl

2.157f

158.7f

155.2f

114.4f

155.44f

168.0

139.0

118.3

151.92

33.76

21.04

4.50

NM

111.8

223.0

129.9

185.90

-321.75

-1.71



FM

168.3

164.1

116.2

165.52

43.62

76.39

3.80

AF Ni 2 FeAl

169.13b 176.92f

162.88f

123.44f

164.23f

188.8

154.5

110.8

164.43

30.81

20.49

5.34

NM

157.8

196.4

138.2

183.53

-46.92

-2.31



FM

122.7

195.6

123.6

171.31

-153.80

-0.58



-71.26

-1.23



AF Ni 2 CoAl

184.42b AF a

141.1

186.8

123.5

170.45

Ref. [9]; b Ref. [3]; c Ref. [17]; d Ref. [16]; e Ref. [6]; f Ref. [7].

3.3. Mechanical stability We also present the calculation of elastic constants of the Heusler alloys for the L2 1 , and L1 0 structure and the results in the unit of GPa are listed in Table 3 and Table 4, respectively. The stability conditions of the orthorhombic structures are shown[23] to be

Cii > 0, i = 1, 2,3, 4,5, 6;

(4)

Cii + C jj − 2Cij > 0, i, j = 1, 2,3, and i ≠ j;

(5)

C11 + C22 + C33 + 2 (C12 + C13 + C23 ) > 0;

(6)

where it can be reduced to C11 + 2C12 > 0 , C44 > 0, C11 > 0 and C11 − C12 > 0 in the L2 1 cubic structure. Once the elastic tensor Cij is calculated, we can further calculate the compliance tensor sij = Cij−1 , average bulk modulus in the Voigt ( BV ) and Reuss ( BR ) form; shear modulus GV and GR and universal anisotropy index AU [24] from the formula

BV =

C11 + C22 + C33 + 2 (C12 + C13 + C23 ) ; 9

1 = s11 + s22 + s33 + 2 ( s12 + s13 + s23 ); BR 15×GV = C11 + C22 + C33 − (C12 + C13 + C23 ) +3(C44 + C55 + C66 ); 15 = 4 ( s11 + s22 + s33 ) − 4 ( s12 + s13 + s23 ) GR

(7) (8)

(9)

(10)

+3( s44 + s55 + s66 ); AU =

5GV BV + − 6 ≥ 0; GR BR

(11)

Table 4: Calculated elastic constant C11 , C12 , C13 , C33 , C44 , C66 , average bulk modulus B = (BV+ BR)/2, and average shear modulus G = (GV+ GR)/2 (in GPa) in the L1 0 crystal structure.

L1 0 crystal structure C 11

C 12

C 13

C 33

C 44

C 66

B

G

GPa Ni 2 VAl NM

296.1

76.7

168.1 205.5

15.6

109.7 180.4 233.1

NM

300.2

72.7

187.0 187.2

-0.1

113.7 187.1 193.1

FM

283.2

44.0

150.6 174.9

11.8

119.3 159.1 236.2

AF

235.8

83.2

151.1 163.8

54.1

123.3 156.2 168.1

NM

235.3 157.8 167.8 240.3

91.9

115.3 187.3 182.1

FM

268.7

41.1

141.8 168.2

13.7

112.3 150.5 228.3

AF

267.1

49.3

138.6 194.1

15.5

118.8 153.3 245.3

NM

352.4

43.7

154.3 278.1 -29.4 114.4 187.1 651.7

FM

277.8

50.1

162.0 165.7

1.4

111.8 163.3 198.1

AF

304.9

55.9

132.7 241.1

3.6

105.3 165.9 293.9

NM

335.4

37.3

170.7 224.3 -29.4 127.7 183.2 359.0

FM

320.5

30.7

152.8 238.1 -23.4 106.4 171.6 516.4

AF

323.8

38.2

143.3 259.2 -24.8 109.4 172.4 853.4

Ni 2 CrAl

Ni 2 MnAl

Ni 2 FeAl

Ni 2 CoAl

Calculated elastic constants, bulk modulus, shear modulus and universal elastic anisotropy of the Ni 2 XAl in the L2 1 structures are listed in Table 3. We found that in the L2 1 structure, Ni 2 MnAl (NM), Ni 2 FeAl (NM) and Ni 2 CoAl (NM, FM, and AF) are unstable because the stability criterion C 11 -C 12

> 0 , as well as AU ≥ 0 , are not satisfied. The universal anisotropy index AU for isotropic crystals

is zero. A larger universal anisotropy index indicates a stronger crystalline anisotropy. In among stabled Ni 2 XAl Heusler alloys, Ni 2 FeAl in the ferromagnetic state has the largest AU . It is seen that in stable Ni 2 XAl, C 11 is lower than C 44 . The elastic constant C 11 represents stiffness against principle strains and C 44 represents shear deformation. Furthermore, The bulk modulus represents a measure of fracture stress and the shear modulus represents a measure of resistance for dislocation. In particular, the bulk modulus to shear modulus ratio ( B / G ) is a useful quantity suggested by Pugh[25] for ductility or brittleness. The critical B / G value for determining the ductile or brittle material is B / G ∼ 1.75[25]. Very interestingly, all magnetic Ni 2 XAl alloys are ductile materials in the L2 1 structure. Pettifor[26] also suggests that the ductility or brittleness can be determined from Cauchy pressure in metals and intermetallics. For cubic structure, the Cauchy pressure is defined as C 12 -C 44 and it is positive for metallic bonding and negative for directional bonding. In the L2 1 Ni 2

XAl alloys, calculated Cauchy pressure is all positive which also implies ductility. In orthorhombic crystals, the Cauchy pressure is defined by C 12 -C 66 , C 13 -C 55 , and C 23 -C 44 . From Table 4, it is seen that in the Ni 2 VAl, the calculated elastic constants in the L2 1 and L1 0 phase are identical. For example, calculated C 11 in the L2 1 structure of Ni 2 VAl is 205.3 GPa whilst it is 296.1 GPa in the L1 0 structure. This difference in the ratio (296.1/205.3) corresponds to the ratio in the lattice constant (5.793/4.10). Further, C 33 of Ni 2 VAl in the L1 0 structure is identical to C 11 in the L2 1 structure. In the L1 0 structure, the stable criteria are listed in Eq. 3, Eq. 4 and Eq. 5. Clearly, in Table 4, the C 44 is negative in nonmagnetic Ni 2 CrAl and Ni 2 FeAl and nonmagnetic, ferromagnetic and antiferromagnetic Ni 2 CoAl, indicating these materials are mechanically unstable. The B / G ratio for the L1 0 Ni 2 XAl alloys is all lower than 1.75, indicating a brittleness behavior for these materials in the L1 0 structure.

4. Conclusions We’ve calculated Ni-based Ni 2 XAl (X= V, Cr, Mn, Fe, and Co) Heusler alloys for both L2 1 and L1 0 structures under the framework of density functional theory. We found that in the L2 1 crystal structure, the ferromagnetic state is the ground state, except Ni 2 VAl. Ni 2 MnAl has the lowest formation energy as well as magnetization energy. However, the magnetization energy differences between FM and AF states are small. In the L1 0 tetragonal phase, we found that Ni 2 CrAl and Ni 2 CoAl are energetically favorable in the antiferromagnetic state. To study the stability between L2 1 and L1 0 crystal structure, we calculate the cohesive energy. We found that Ni 2 VAl in the NM state has the largest cohesive energy. In the FM state, Ni 2 CrAl has the lowest cohesive energy. We also calculate the elastic constants to study the mechanical stability.

5. Acknowledgment The authors acknowledge supports from the China Medical University (CMU107-N-32) and Ministry of Science and Technology (MOST-107-2112-M-039) . They also thank the Academia Sinica of the ROC and the NCHC of Taiwan for providing CPU time.

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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: