Half-metallicity and magnetism of Heusler alloys Co2HfZ (Z = Al, Ga, Ge, Sn)

Accepted Manuscript Research articles Half-metallicity and magnetism of Heusler alloys Co2HfZ (Z=Al, Ga, Ge, Sn) Babiker A. Salma, Guoying Gao, Kailun Yao PII: DOI: Reference:

S0304-8853(17)30115-4 http://dx.doi.org/10.1016/j.jmmm.2017.04.099 MAGMA 62730

To appear in:

Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

13 January 2017 5 March 2017 15 April 2017

Please cite this article as: B.A. Salma, G. Gao, K. Yao, Half-metallicity and magnetism of Heusler alloys Co2HfZ (Z=Al, Ga, Ge, Sn), Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm. 2017.04.099

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Half-metallicity and magnetism of Heusler alloys Co2HfZ (Z=Al, Ga, Ge, Sn) Salma Babiker A1,2, Guoying Gao 1,* and Kailun Yao1,** 1

School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China 2

Department of Physics, Faculty of Education, University of Al Fashir, Sudan

Abstract The electronic and magnetic properties of novel Heusler alloys Co2HfZ (Z = Al, Ga, Ge, Sn) with Hg2CuTi- and Cu2MnAl-type structures are investigated by first-principles calculations. It is found that the Cu2MnAl-type structure is more stable than the Hg2CuTi-type structure for all the Co 2HfZ alloys. The Hg2CuTi-type structures of Co2HfZ show a metallic ferrimagnetic character with a high spin polarization. The Cu2MnAl-type structures have half-metallic ferrimagnetic behavior with minority-spin gaps of 0.38, 0.06, 0.55 and 0.46 eV for Co 2HfAl, Co2HfGa, Co2HfGe and Co2HfSn, respectively. The total magnetic moments of the Cu2MnAl-type alloys are 1µ B, 1µ B, 2µ B and 2µ B, which obey the Slater–Pauling rule (M

tot

= Z

tot

−24). Analysis of the projected density of states indicates clear gaps

resulting from the hybridization of Co (A) and Co (B) atoms. The half-metallicity makes Co 2HfZ with Cu 2MnAl-type structure promising candidates for spintronic applications.

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (G. Gao), [email protected] (K. Yao).

1

I. Introduction Since the discovery of Heusler alloys by Fritz Heusler in 1903, the half-metallicity of Heusler alloys has drawn interest because this characteristic may have applications to electrodes in magnetic tunnel junctions [1]. Heusler alloys feature majority-spin electrons that have normal metallic behavior, while minority spin electrons are semiconducting [2, 3]. Thus, these compounds feature spin polarization around the Fermi level of 100%. Various types of half-metallic compounds have been identified including Heusler alloys, ferromagnetic metallic oxides, zinc-blend binary transition-metal pnictides, and dilute magnetic semiconductors. Among these compounds, Heusler alloys are extremely important, particularly for potential applications in spintronics, because of their high Curie temperatures and magnetic moments [4, 5]. The first research into the full Heusler alloys, ferromagnetic Cu 2MnAl and Cu2MnSn, described the compounds’ magnetic behavior in terms of chemical ordering [6, 7]. Heusler alloys are classified as ternary inter-metallic compounds with the general formula X2YZ, where X and Y are transition elements and Z is a main group III or group IV element [8]. Two types of different atoms may be found occupying tetrahedral coordination sites in the crystal lattice. These positions are denoted by Wyckoff coordination as A(0,0,0), B(1/4,1/4,1/4), C(1/2,1/2,1/2) and D(3/4,3/4,3/4). Full Heusler alloys X2YZ have two crystal structure types. The Hg2CuTi-type structure has the space group symmetry (F-43m, no.216), and the number of 3d electrons of the Y atom is greater than that of the X atom. This compound is known as the inverse Heusler structure [9], in which the two X atoms occupy the A and B sites. The second structural type Cu2MnAl has the space group (Fm-3m, no. 225) and contains two X atoms occupying the A and C sites. Carbonari et al. [10] has reported experimental results showing the presence of Co2HfZ in Co 2YZ (Y = Sc, Ti, Hf, V, Nb, Cr; and Z = Al, Ga, Si, Ge, and Sn). Recently, Yin et al. [11] explained that the structure of Co2HfZ (Z = Al, Ga, and Sn) can be attributed to the high melting point of Hf. Theoretical work by Rauf et al. [12] determined the half-metallic band structure in Co 2HfSn. However, there have been no 2

reports on the electronic and magnetic properties of Co2HfZ except for Co2HfSn. In this study, we use the first-principles plane-wave full-potential method to address the structural, electronic and magnetic properties of full Heusler alloys Co2HfZ (Z = Al, Ga, Ge, Sn). The crystal structures are described by four interpenetrating

face-centered-cubic

(fcc)

sub-lattices.

Knowledge

of

these

compounds is particularly important for advancing spin-transfer torque applications [13]. The main goal of this work is to predict the possible half-metallicity in Heusler compounds Co2HfZ (Z = Al, Ga, Ge, Sn) with Hg2CuTi- and Cu2MnAl-type structures.

2. Computational method To study the structural, electronic and magnetic properties of bulk Co2HfZ (Z = Al, Ga, Ge, Sn) full Heusler alloys, we use the first-principles calculations with the full potential linear augmented plane wave method, implemented in the WIEN2K code [14]. This method has been used to study various half-metallic systems in our previous works [15-18]. In our calculations, we adopt the exchange and correlation energy using the Predew–Burke–Ernzeh (PBE) functional of the generalized gradient approximation (GGA) [19]. The linear muffin-tin orbital (LMTO) method for the atomic spherical potential and charge density was expanded to l = 10 in spherical harmonic functions inside the muffin-tin sphere. The values of the atomic sphere muffin-tin radius RMT are given in atomic units (au), in Table 1. In these calculations, we set the charge convergence as 0.0001e. The kinetic energy cut-off of RmtKmax equals to 8.0, where Rmt is the smallest atomic sphere radius and Kmax is the largest reciprocal lattice vector making the plane wave expansion. Brillouin zone integration was performed with the 14×14×14 k-mesh. The total energy difference between succeeding iterations during the self-consistency cycles is set to 10−5 Ry/Cell.

3. Results and discussion 3.1. Electronic band structure and density of states at equilibrium lattice constant 3

We consider two structural types of Hg2CuTi and Cu2MnAl for full Heusler alloys Co 2HfZ (Z = Al, Ga, Ge, Sn). The equilibrium structural parameters of these compounds are presented in Table 1, containing the optimized bulk modulus (B0), a pressure derivative of bulk modulus (BP), lattice constants (a) and the minimum value of total energy (E0). These parameters are evaluated by fitting Mumaghan equation of state (EOS), which is essential in studying the high pressure [20, 21]. Table 1 indicates that the Cu2MnAl-type structure is more stable in energy than the Hg2CuTi-type structure for all the Co 2HfZ alloys. This is reasonable, because the valence electron number of Hf atom is smaller than that of Co atom. We also note that the lattice constants of Cu2MnAl-type structure are in good agreement with reported experimental values [22, 23]. Moreover, we further evaluate the formation enthalpy (∆H) for ternary traditional metal by using the Miedema theory [24, 25]. The formation enthalpy of compounds Co2HfZ (Z=Al, Ga, Ge, Sn) has been calculated by: ∆H = ECo2HfZ − 2ECo − EHf − EZ

(1)

where ∆H is the standard formation enthalpy, and ECo2HfZ, ECo, EHf and EZ are the total energies of compounds Co 2HfZ, atoms Co, Hf and Z, respectively. The negative ∆H shown in Table 1 indicates the structural stability of Cu2MnAl-type structure. The electronic band structures for alloys Co2HfZ (Z = Al, Ga, Ge, Sn) in the irreducible Brillouin zone are shown in Fig. 1. These findings demonstrate the differences between Hg2CuTi and Cu2MnAl structures in terms of the majority (spin-up) and minority (spin-down) channels. The band structures of all four alloys with the Hg2CuTi-structure show a metallic behavior, because both majority and minority spin electrons cross the Fermi level. Differently, for the Cu2MnAl-type structure, all four alloys show metallic characteristic in the majority spin channel, while there is an energy gap around the Fermi level for the minority spin channel, and thus all four alloys with Cu2MnAl-type structure are half-metals. Note that the half-metallicity is weak for Co2HfGa, because the minority-spin valence bands cross the Fermi level a little. The energy gaps in the minority spin channels are about 0.38, 0.06, 0.55 and 0.46 eV for Co2HfAl, Co2HfGa, Co2HfGe and Co 2HfSn, respectively. Table 2 presents the spin flip gap, which corresponds to a minimum energy of the valence band required to flip a minority spin electron from the majority channel Fermi 4

level (Esf). The non-zero spin flip gaps [26] imply that these compounds studied are true half-metals. Hybridization between d-d orbits, indicated in Fig. 2, assumes bonding and anti-bonding states derived from splitting of the 3d electron in a cubic crystal field [27]. Contributions from Co(1), Co(2) and Hf atoms generate the bonding and anti-bonding states, which lead to the band gap. Fig. 2 (left side) depicts the interaction between two neighboring Co atoms (different from each other). Fig. 2 (right side) shows five Co orbitals, including doublet e g and triplet t 2 g , which coupled with the Hf atom d xy , d yz , d xz , d z 2 and d x 2 − y 2 orbitals. Below the Fermi level the degenerate bonding e g and t 2 g orbitals, and a non-bonding triplet t1u , are localized on the higher-valence transition metal [28]. Above the Fermi level anti-bonding states include the unoccupied e g and t 2 g orbitals and non-bonding eu , which are localized on the lower-valence transition metal and have high energy. The eu and t 2 g orbitals are not coupled to the Hf atom, leaving a gap between the bonding

t1u and anti-bonding eu levels, i.e., a gap appears in the projected density of states of

the minority spin channel for all four compounds. Fig. 3 shows the total and projected density of states for the compounds Co2HfAl with Hg2CuTi- and Cu2MnAl-type structures. These results can be explained by Co(1) and Co(2) being in different environments. As shown in Fig. 3, Co2HfAl with the Hg2CuTi-type structure has a minority spin state that cuts the Fermi level, meaning that the Fermi level falls into the minority states, which limits the half-metallic character. Accordingly, the projected density of states at the Fermi level shifts toward the empty bonding region and leads to peaks in the density of states for Co(1)-d e g and Co(1)-d t 2 g . The Co(2)-d e g and Co(2)-d t 2 g states above and below the Fermi level featured a higher projected density of states than those of Co(1). For the case of the Cu2MnAl-type structure, Co(1)-d t 2 g and Co(2)-d t 2 g have large energy gaps and show strong hybrid crystal behavior. The minority-spin down and majority-spin up bonding levels have a high density of states below the Fermi level, as expected from the 3d-states, Co(1)-d t 2 g and Co(2)-d t 2 g . The anti-bonding levels represent minority

5

spin electrons that are concentrated at Co(1)-d e g and Co(2)-d e g . The 5d states of the Hf-d e g and Hf-d t 2 g atoms are mainly located above the Fermi level with a small gap in the projected density of states predicted below the Fermi level. The results shown in Fig. 3 for Co2HfAl reflect contributions to the density of states from spin up and spin down electrons around the Fermi level, from −1 to −2.5 eV, that are related to 3d orbitals of Co(1), Co(2) atoms and the 5d orbitals of Hf atoms. The energy bands from −5 to −2.5 eV are attributed to p orbitals of Z (Al, Ga, Ge, and Sn) atoms and those from −6 to −10 eV are the lowest bands, arising from the s orbitals of the Z atom. The lower orbital energy values for Z are related to the effective nuclear charge of the Z atom, indicating weaker hybridization between the Co, Hf and Z atom [28-31]. The results of projected density of states for compounds Co2HfGa, Co 2HfGe and Co2HfSn (not shown in Fig. 3) are similar to those of Co2HfAl. 3. 2. Magnetic properties We calculate the interstitial, total and partial spin magnetic moments in the compounds Co2HfZ (Z = Al, Ga, Ge, Sn). Values for the Hg2 CuTi- and Cu2MnAl- type structures are shown in Table 2. The Cu2MnAl-type structures, give near integral values magnetic moments of 0.999µ B, 1.002µ B 1.999µ B and 1.998µ B for the Co 2HfAl, Co2HfGa, Co 2HfGe, Co 2HfSn compounds, respectively. This indicates that all the compounds have half-metallic-features. The calculated magnetic moment of Co2HfSn agreed well with the previously reported value [32]. Notably, the compounds Co 2HfZ (Z = Al, Ga, Ge, Sn), which features the Hg2CuTi-type structure, did not show integral values confirming that these compounds were not half-metals. The total number of valence electrons in Co2HfAl, Co 2HfGa, Co2HfGe, and Co2HfSn, in a unit cell of the Cu2MnAl-type structure are 25, 25, 26 and 26, respectively. The magnetic moments show a linear relationship with the number of valance electrons indicating Slater–Pauling behavior in the full Heusler alloys, as given by Equation (2): Mtot = Ztot − 24

(2)

where Mtot is the total spin magnetic moment in µ B and Z

tot

is the total number of

valence electrons. Equation (2) gives the total magnetic moments as 1, 1, 2, and 2 µ B, 6

for Z = Al, Ga, Ge and Sn, respectively. The total magnetic moment of Co(A) and Co(B) includes contributions from differences in their atomic environments leading to some variations, as observed for the Hg2CuTi and Cu2MnAl-structures. Furthermore, our analysis of the magnetic interactions show anti-parallel coupling between Co(A) and Co(B) to Hf and Z atoms, indicating ferrimagnetic ordering [33, 34] in Heusler compounds Co2HfAl and Co2HfGa and ferromagnetic ordering (parallel coupling) in Co2HfGe and Co2HfSn. The spin moment of the Z atoms is always negligible and has a small effect on the total spin magnetic moment. 3.3. Spin Polarization The absolute value of the spin polarization P of the total density of states for all compounds can be given by Equation (3) [35]: P=

N ↑ (E f ) − N ↓ (E f )

%

(3)

N ↑ (E f ) + N ↓ ( E f )

where N↑(Ef) and N↓(Ef) are the carrier density of states in the majority and minority channels at the Fermi level, respectively [36]. For Co2HfZ (Z = Al, Ga, Ge, Sn) with the Hg2CuTi-type structure, although they are metals, their spin polarizations 80.37%, 74.29%, 75.63% and 69.43% respectively, are still high. For the case of the Cu2MnAl-type structure, we confirm that the spin down N↓(Ef) for all four compounds equal to zero, and thus the spin polarization of electrons at the Fermi level are 100%, as shown in Table 3.

4. Conclusion We have used the first-principles method to calculate electronic and magnetic properties of Co2HfZ (Z = Al, Ga, Ge, Sn) with Hg2CuTi and Cu2MnAl-type structures. We find that the Cu 2MnAl-type structure is more stable than the Hg2CuTi-type

structure.

The

Cu2MnAl-type

structures

show

half-metallic

ferrimagnetic behavior with the magnetic moments of 1µ B, 1µ B, 2µ B and 2µ B, respectively, having 100% spin polarization at the Fermi level. These studies indicate that Cu2MnAl-type Heusler alloys of Co2HfZ would be useful in spintronic

7

applications.

Acknowledgements This work was funded by the China National Natural Science Foundation under Grant No. 11274130 and 11474113.

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[12] S. Rauf, S. Arif, M. Haneef, B. Amin, J. Phys. Chem. Solids 76 (2015) 153 [13] G. Chris, V.d. Walle, J. Neugebauer, J. Appl. Phys. 95 (2004) 3851. [14] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, J. Luitz, wien2k, An augmented plane wave + local orbitalsprogram for calculating crystal properties, Vienna Uni of Tech, Austria (2001). [15] G.Y. Gao, K.L. Yao, E. Sasioglu, L.M. Sandratskii, Z.L. Liu, J.L. Jiang, Phys. Rev. B 75 (2007) 174442. [16] G.Y. Gao, L. Hu, K.L. Yao, B. Luo, N. Liu, J. Alloys Compd. 551 (2013) 539. [17] L. Xiong, L. Yi, G.Y. Gao, J. Magn. Magn. Mater. 360 (2014) 98. [18] G. Gao, G. Ding, J. Li, K. Yao, M. Wu, M. Qian, Nanoscale 8 (2016) 8986. [19] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [20] F.D. Murnaghan, Phys. 30 (1944). [21] R.S. Chauhan, K. Lal, C.P. Singh, Indian J. Phys. 85 (2011) 1341. [22] R. da Silva, R.N. Saxena, J. Schaf, F.P. Livi, F.C. Zawislak, Hyperfine Interact. 9 (1981) 489. [23] S.D. De Souzai, R.N. Saxena, Hyperfine Interact. 34 (1987) 431. [24] W.C. Wang, J.H. Li, H.F. Yan, B.X. Liu, Scripta Mater. 56 (2007) 975. [25] M.K. Hussain, G.Y. Gao, K.L. Yao, J. Supercond. Nov. Magn. (2015). [26] J. Nehra, N. Lakshmi, K. Venugopalan, Physica B 459 (2015) 46. [27] C. Felser, A. Hirohata, Heusler alloys, properties, growth, applications. Springer Series in Materials Sciences 222 (2015). [28] I. Galanakis, P.H. Dederichs, Phys. Rev. B 66 (2002) 134428. [29] Y.J. Zhang, W.H. Wang, H.G. Zhang, 420 (2013) 86.

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10

Tables

Table 1 The adopted values of RMT (in a.u.), the optimized lattice parameter a in (Å), formation enthalpy ∆H (in eV), bulk modulus B in (GPa), derivative of bulk modulus BP and total energy E0 in (RY) for Co2HfZ (Z = Al, Ga, Ge, Sn) with Hg2CuTi- and Cu2MnAl-type structures (labeled as (1) and (2), respectively).

Compounds

Co

Hf

Z

Co2 HfAl (1)

2.48

2.48

2.33

Co2 HfAl (2)

2.46

2.46

Co2 HfGa (1)

2.48

Co2 HfGa (2)

a (Å)

∆H

B(GPa)

BP

6.104

−0.48

151.890

4.541

−36255.223

2.31

6.041

−2.12

177.452

4.610

−36255.378

2.48

2.33

6.107

−0.25

161.956

4.851

−39657.827

2.45

2.45

2.30

6.025

−1.45

185.711

5.390

−39657.903

Co2 HfGe (1)

2.48

2.48

2.33

6.102

−0.29

166.567

5.439

−39967.772

Co2 HfGe (2)

2.45

2.45

2.36

6.025

−1.39

183.705

4.766

−39967.876

Co2 HfSn (1)

2.50

2.50

2.42

6.333

149.776

4.848

−48127.730

Co2 HfSn (2)

2.50

2.50

2.38

6.230

164.471

4.040

−48127.881

11

0.42 −1.45

E0

Table 2 The calculated total and atomic spin magnetic moments (in µ B), energy gap (Eg), spin-flip gaps (Esf) and energy difference between two structural types (∆E) in the primitive cell for Co2HfZ (Z = Al, Ga, Ge, Sn) with Hg2CuTi- and Cu2MnAl-type structures. Compounds Structure

Co2HfAl (1)

Co2HfGa (2)

(1)

Co2HfGe (2)

(1)

Co2HfSn

(2)

(1)

(2)

M int erstitial

−0.072

−0.113

−0.079

−0.101

−0.104

−0.083

−0.111

−0.098

Mtot

2.539

0.999

2.478

1.002

2.259

1.999

1.884

1.998

M Co( A)

1.179

0.610

1.102

0.601

1.111

1.058

0.617

1.076

M Co(B)

1.490

0.610

1.521

0.601

1.321

1.058

1.487

1.076

M Hf

−0.060

−0.090

−0.080

−0.091

−0.099

− 0.073

−0.128

−0.073

MZ

0.001

−0.016

0.015

−0.007

−0.031

0.039

0.019

0.017

Eg (eV)

-

0.38

-

0.06

-

0.55

-

0.46

Esf (eV)

-

0.21

-

0.11

-

0.38

-

0.27

1.890

∆E (eV)

1.446

0.104

0.147

Table 3 The density of states N↑for majority, N↓for minority electrons and spin polarization (P) at the Fermi level for Co2HfZ (Z = Al, Ga, Ge, Sn) with Hg2CuTi (1) and Cu2MnAl (2)-type structures. Compounds

Co2HfAl(1)

Co2HfAl(2)

N↑ (Ef)

0.74

1.28

0.82

1.34

0.92

N↓ (Ef)

6.80

0.00

5.56

0.00

80.37

100

74.29

100

P (%)

Co2HfGa(1)

Co2HfGa (2)

12

Co2HfGe (1)

Co2HfGe (2)

Co2HfSn (1)

Co2HfSn (2)

0.77

1.85

0.85

6.63

0.00

0.00

0.00

75.63

100

69.43

100

Figures and Figure Captions 4

(b)

2

2

0

0 Energy(eV)

Energy(eV)

(a) 4

-2 -4

-6

-8

-8

-10 Λ

Γ

∆

X

-10 K (d) 4 W 2

0

0 Energy(eV)

Energy(eV)

L

2

-2 -4

-8

-8

W

L

Λ

Γ

∆

X

-10 K W (f) 4 2

0

0 Energy(eV)

2

-2 -4

-8

-8

Λ

Γ

∆

X

-10 K 4W (h)

2

0

0 Energy(eV)

2

-2 -4

X

K

L

Λ

Γ

∆

X

K

L

Λ

Γ

∆

X

K

L

Λ

Γ

∆

X

K

-2 -4

-6

-6

-8

-8

-10 W

∆

-4 -6

L

Γ

-2

-6

-10 (g)4 W

Λ

-4 -6

-10

L

-2

-6

(e) 4

Energy(eV)

-4

-6

(c) 4 W

Energy(eV)

-2

-10

L

Λ

Γ

∆

X

K

W

Fig. 1. Band structures in the majority (black line) and minority spin channels (red line) of Co2HfAl (a and b), Co2 HfGa (c and d), Co2HfGe (e and f) and Co2 HfSn (g and h) for the Hg2CuTi- (left part) and Cu2 MnAl- (right part) type structures, respectively. The dashed dot line at the zero energy axes corresponds to the Fermi level. 13

Anti-bonding states 2eg 3t2g

Co (1)

Co (2) 2eg 3teg

2eu 3t1u 3t2g 2eg

Hf

EF

dz2, dx2-y2 d xy,dyz,dxz

Co (1)-Co (2) Bonding states

Fig. 2. Orbital hybridization between Co(1)-Co(2) (left part) and between Co-Hf (right part) atoms in Co2HfZ (Al, Ga, Ge, Sn).

14

Hg2CuTi

5 Co2HfAl-total 0 -5 Co1-tot 2 Co1-deg 0 Co1-dt2g -2 Co2-tot 2 Co2-deg 0 Co2-dt2g -2 2 Hf-tot 1 Hf-deg 0 Hf-dt2g -1 -2 Al-tot 0.3

spin-up

Density of States (States/eV)

spin-dn

Al-S Al-P

0.0 -0.3 -9

-6

-3

0

3

6

Cu2MnAl

Density of States (States/eV)

5 Co2HfAl-total 0 -5 Co1-tot Co1-deg Co1-dt2g

2 0 -2 2 0 -2 2 1 0 -1 -2 0.3 0.0 -0.3

Co2-tot Co2-deg Co2-dt2g Hf-tot Hf-deg Hf-dt2g Al-tot Al-S Al-P

-9

-6

-3

0

3

6

Energy(eV) Fig. 3. Total and projected spin-up (positive) and spin-down (negative) density of states for Co2HfAl with Hg2CuTi- and Cu2MnAl-type structures. The dashed line means the Fermi level at 0 eV. 15

Highlights

1. The Cu2 MnAl-type structure is more stable than the Hg2CuTi-type structure for Co2HfZ alloys. 2. The Cu2MnAl-type structures of Co2 HfZ exhibit half-metallic ferrimagnetic behavior. 3. The total magnetic moments of the Cu2MnAl-type Co2 HfZ obey the Slater–Pauling rule (Mtot =Ztot −24).

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