ARTICLE IN PRESS
Physica B 403 (2008) 200–206
www.elsevier.com/locate/physb
Ab-initio investigation of electronic properties and
magnetism of half-Heusler alloys XCrAl (X ¼ Fe, Co, Ni)
and NiCrZ (Z ¼ Al, Ga, In)
Hongzhi Luoa,, Zhiyong Zhua, Guodong Liua, Shifeng Xua, Guangheng Wua,
Heyan Liub, Jingping Qub, Yangxian Lib
a
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, PR China
b
School of Material Science and Engineering, Hebei University of Technology, Tianjin 300130, PR China
Received 16 June 2007; received in revised form 27 August 2007; accepted 27 August 2007
Abstract
The electronic structures and magnetism of the half-Heusler alloys XCrAl (X ¼ Fe, Co, Ni) and NiCrZ (Z ¼ Al, Ga, In) have been
investigated to search for new candidate half-metallic materials. Here, we predict that NiCrAl, and NiCrGa and NiCrIn are possible halfmetals with an energy gap in the minority spin and a completely spin polarization at the Fermi level. The energy gap can be attributed to
the covalent hybridization between the d states of the Ni and Cr atoms, which leads to the formation of bonding and antibonding peaks
with a gap in between them. Their total magnetic moments are 1mB per unit cell; agree with the Slater–Pauling rule. The partial moment
of Cr is largest in NiCrZ alloys and moments of Ni and Al are in antiferromagnetic alignment with Cr. Meanwhile, it is also found that
FeCrAl is a normal ferromagnetic metal with a magnetic moment of 0.25mB per unit cell and CoCrAl is a semi-metal and non-magnetic.
r 2007 Elsevier B.V. All rights reserved.
PACS: 71.20.Be; 71.20.Lp; 75.50.Cc
Keywords: Half-Heusler alloy; Band structure; Half-metallicity
1. Introduction
The rapid development of spintronics [1,2] gets much
attention in recent years. Widely possible applications such
as single spin electron sources and spin injectors have been
envisaged [3,4]. One key ingredient for spintronics is the
source of the spin-polarized charge carriers. An ideal
choice is the half-metallic materials, including half-metallic
ferromagnets (HMFs) and half-metallic antiferromagnets.
All the half-metallic materials have a common character in
their energy bands, they are semiconductor-like in one spin
direction at the Fermi level whereas the other spin is
strongly metallic, which results in a complete (100%) spin
polarization of the conduction electrons at the Fermi level.
The HMFs were first discovered by theoretical band
Corresponding author. Tel.: +86 10 8264 9247; fax: +86 10 6256 9068.
E-mail address: [email protected] (H. Luo).
0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.physb.2007.08.214
structure calculations for half-Heusler alloy NiMnSb [5].
Then much investigation has been put on both half- and
full-Heusler alloys [5–14]. For the so-called half-metallic
antiferromagnet, its magnetic properties are unique. It has
ferrimagnetic coupling with completely compensated magnetic moment and 100% spin polarization at the Fermi
level. So, it has an advantage for some technical applications, for it is non-magnetic and not easy to be affected by
external magnetic fields [15–17]. In 1991, de Groot [15]
predicted MnCrSb with half-Heusler alloy structure to be a
half-metallic antiferromagnet. Later, similar Heusler alloy
Fe8MnV7Sb7In [17] and Mn3Ga [16] has also been
predicted by band structure calculation. However, these
alloys are difficult to be synthesized with its stoichiometric
form.
Recently, Wurmehl et al. [16] proposed the rule for
designing half-metallic antiferromagnets that is this kind of
material might be expected in half-Heusler alloys with 18
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H. Luo et al. / Physica B 403 (2008) 200–206
valence electrons or full-Heusler alloys with 24 electrons
and with Mn on the Y site. The Mn at Y site in Heusler
alloys tends to have a high, localized magnetic moment and
cancels the moments at X sites. In fact, a large magnetic
moment has also been found in Cr at Y site [18]. So, it is
possible to design a half-metallic antiferromagnet in halfHeusler alloys XYZ with 18 valence electrons and Cr atom
at Y site. Here, we select half-Heusler alloy CoCrAl to find
if half-metallicity with compensated magnetic moment can
be expected in it as has been found in MnCrSb [15].
Meanwhile, it is known that the magnetic and electronic
properties of half-Heusler alloy strongly depend on the
number of valence electrons [12]. So, in this paper we
investigate the electronic structure and magnetism of halfHeusler alloys with 17–19 electrons namely XCrAl
(X ¼ Fe, Co, Ni) and NiCrZ (Z ¼ Al, Ga, In) with firstprinciples band structure calculations. Three possible halfmetals are predicted as NiCrAl, NiCrGa and NiCrIn. A
large magnetic moment on Cr site and antiferromagnetism
coupling between the moments Ni and Cr is observed.
2. Computational methods
We carried out the electronic structure calculation using
the self-consistent full-potential linearized-augmented
plane wave (FLAPW) method based on the local spindensity approximation within the density functional theory
[19], where the potential and/or the charge density in the
crystal are treated with no shape approximation. One
hundred and eighty-two k points are employed in the
irreducible Brillouin zone. The self-consistent calculation
stops as the charge density deviation is less than 0.01 me/
a.u. and the total energy deviation is better than 0.1 mRy
per cell. The density plane-wave cutoff is Rkmax ¼ 8.0. The
electron states were treated in a scalar relativistic approximation. Using the energy eigenvalues and eigenvectors at
these points, the density of states (DOS) was determined by
the tetrahedral integration method [20].
The half-Heusler alloy XYZ crystallizes in the facecentered cubic (fcc) structure with one formula unit per
primitive cell. The space group is F4-3m. In this study, Fe,
Co and Ni represent the X, while Cr and (Al, Ga, In)
represent the Y and Z atom, respectively. The site
preference in half-Heusler alloys has been studied
201
[13,21,22]. It is found that the conventional stable structure
is that the Y and Z atoms locate at (0, 0, 0) and (12; 12; 12) sites
and form the rock salt structure while the X atom locates in
the octahedrally coordinated pocket, at one of the cube
center site (14; 14; 14) leaving the other site (34; 34; 34) unoccupied.
The crystal with different configurations as XYZ, ZYX and
YXZ has been investigated and the configuration with
more valence electrons at X site is found to be lowest
in energy [13]. In XYZ structure, the X atom has
four Y and four Z as nearest neighbors whereas Y and Z
atoms only have four X atoms in their nearest neighbors
coordinations.
3. Results and discussion
To determine the theoretical lattice parameter, we
perform total energy calculations on XCrAl and NiCrZ
alloys both for the non-magnetic (PM) and the ferromagnetic (FM) states at different lattice parameters.
The calculated equilibrium lattice parameters are listed in
Table 1.
For XCrAl, it is found that when the X atoms belong to
the different column of the periodic table their magnetic
behavior is different. For FeCrAl or NiCrAl, it can be seen
that the ferromagnetic state is more stable in energy than
the paramagnetic one. Whereas in CoCrAl, the total energy
difference between the paramagnetic state and corresponding ferromagnetic one is very small and can be neglected.
Both of them give a zero total and partial magnetic
moment indicating paramagnetic configuration is stable.
The energy difference DE between the FM and PM states
are also listed in Table 1. In NiCrZ alloys, calculations also
indicate the ferromagnetic state is more stable.
In order to get a deep study on the electronic structure
for the XCrAl (X ¼ Fe, Co, Ni) alloys, which have 17, 18
and 19 valence electrons, respectively, we study their
total DOS in paramagnetic configuration first. As shown
in Fig. 1, their shapes are similar and the characters can be
described as follows: the states below 6 eV are mainly s
electrons of Al atom, which are relatively small and
separated from the d states by a dip in DOS. The low
energy part around 5 eV are mainly the p states of Al
atom in the occupied valence states, which hybridize with
p and d electrons of the X atoms and determine the
Table 1
The equilibrium lattice constants, total energy differences DE, calculated total and partial magnetic moments, and band gap, HM gap width together with
spin polarization ratio P for the XCrAl (X ¼ Fe, Co, Ni) and NiCrZ (Z ¼ Al, Ga, In) alloys
Compound VEN Lattice constant (Å) DE (eV) Mt (mB) MX (mB) MY (mB) MZ (mB) Gap width (eV) HM gap (eV) P (%) Physical property
FeCrAl
CoCrAl
NiCrAl
NiCrGa
NiCrIn
17
18
19
19
19
5.57
5.52
5.49
5.51
5.70
0.04
0.002
0.20
0.19
0.30
0.25
0.00
1.00
1.00
1.01
0.3
0.00
0.36
0.26
0.32
0.04
0.00
1.60
1.46
1.54
0.00
0.00
0.24
0.20
0.20
VEN, valence electrons number; SM, semi-metal; HMF, half-metallic ferromagnet.
–
–
0.45
0.52
0.51
–
–
0.02
0.17
0.07
11
–
100
100
100
Metal
SM
HMF
HMF
HMF
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H. Luo et al. / Physica B 403 (2008) 200–206
202
20
10
FeCrAl
(17)
CoCrAl
(18)
NiCrAl
(19)
NiCrGa
(19)
NiCrIn
(19)
0
DOS (electrons/eV)
10
0
10
0
20
10
0
20
10
0
-8
-6
-4
-2
0
Energy (eV)
2
4
6
Fig. 1. Total DOS in paramagnetic state for FeCrAl, CoCrAl, NiCrAl, NiCrGa and NiCrIn alloys. The number of valence electrons is shown in the
figure.
occupation degree of the p–d orbitals. The states of 3d
metal atoms extend from 4 to +2 eV and hybridize with
each other. It has been reported that the covalent
hybridization between the lower-energy d states of the
high-valent transition metal atom X and the higher-energy
d states of the lower-valent transition metal Cr can lead to
the formation of bonding and antibonding bands. The
bonding hybrids are localized mainly at the high-valent
transition metal atom site while the unoccupied antibonding states mainly at the lower-valent transition metal site
[23]. So, a d–d band gap is formed near the Fermi level.
There are nine bands below the d–d gap and can
accommodate 18 valence electrons in paramagnetic state
[13]. So, for a half-Heusler alloy with 18 valence electrons
the Fermi level will locate in the energy gap and usually
make it a semiconductor or a semi-metal. When the alloy
has more or fewer than 18 valence electrons, the Fermi
level will be shifted and locate at the antibonding or
bonding peak. The paramagnetic state is no longer stable in
energy when EF lies in the antibonding peak. Spin
polarization and forming magnetic moment can release
this instability [13]. It has been reported that if the
paramagnetic electronic structure supports an energy gap,
then in the process of spin polarization, the half-metallic
ferromagnetism may be stabilized depending on the relative
position of the gap and the Fermi level [24]. The
paramagnetic electronic structure of other HMF systems
like Fe2MnSi [25], NiMnSb, FeVSb [24] also sustains a gap
close to the EF. So, here, we will discuss their spinpolarized DOS below.
The total and partial DOS in ferromagnetic states for
XCrAl alloys are given in Figs. 2–4. It is clear that the three
alloys are different in their magnetic properties. CoCrAl
has 18 valence electrons and in both spin directions the
nine bands are occupied. So, it has a symmetrical DOS in
majority and minority spins and a zero magnetic moment.
In this calculation, CoCrAl is a semi-metal rather than the
wanted half-metallic antiferromagnet. The situation in
NiCrAl and FeCrAl is different; they show ferromagnetic
behavior in their electronic structures.
For NiCrAl, it can be seen from the total DOS that there
is an energy gap in the minority spin at the Fermi level,
whereas there is a high DOS peak in the majority spin.
So, a 100% spin polarization is obtained and NiCrAl
is a half-metal. From the partial DOS, it can be seen
that the Cr atom gains a large exchange energy during the
spin polarization. In the majority spin, the Cr d states
basically lie below the Fermi level and are occupied, which
form a common d band with the Ni d states. While in the
minority spin, the antibonding peak is shifted 1.3 eV above
the Fermi level and unoccupied. The exchange splitting
induces a large magnetic moment on Cr site. The calculated
partial spin moment of Cr is 1.60mB, which is larger than
the calculated total magnetic moment of 1mB and is
compensated by the antiferromagnetic aligned Ni and Al
moments. The exchange splitting is relatively small for Ni d
states and most of the Ni d states are below EF and
occupied. So, the majority and minority spins of Ni d states
are equally populated and form a moment of 0.36mB. The
DOS of Al is almost symmetrical in both spin directions
and only has a small moment of 0.24mB.
The total and partial DOS for FeCrAl alloy are shown in
Fig. 2. It is clear that a magnetic moment is formed in the
spin-polarized calculation. However, there are states at the
Fermi level in both spin directions, so FeCrAl is only a
normal ferromagnetic metal. It can be seen that both the
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DOS (electrons / eV)
H. Luo et al. / Physica B 403 (2008) 200–206
10
5
0
-5
-10
4
2
0
-2
-4
4
2
0
-2
-4
0.8
0.4
0.0
-0.4
-0.8
0.4
0.2
0.0
-0.2
-0.4
203
Total
FeCrAl
Fe
Cr
Al (p)
Al (s)
-8
-6
-4
-2
0
Energy (eV)
2
4
6
DOS (electrons / eV)
Fig. 2. The total and partial DOS in ferromagnetic state for FeCrAl alloy. Minority states are on negative scale.
10
5
0
-5
-10
3
0
-3
-6
3
0
-3
Total
CoCrAl
Co
Cr
0.8
0.4
0.0
-0.4
-0.8
0.4
0.2
0.0
-0.2
-0.4
Al (p)
Al (s)
-8
-6
-4
-2
0
2
4
6
Energy (eV)
Fig. 3. The total and partial DOS in ferromagnetic state for CoCrAl alloy. Minority states are on negative scale.
majority and minority spins are partly occupied. The
exchange splitting of Cr is weaker in FeCrAl than that in
NiCrAl, which introduces a relatively small magnetic
moment of 0.30mB for Cr compared with 1.60mB in NiCrAl.
Nanda and Dasgupta [13] studied the interaction in some
half-Heusler alloys with COHP method. They found that
the X–Y (X ¼ Fe, Co, Ni; Y ¼ Mn, V) interaction is
strongest for Fe and weakest for Ni. This implied a large
admixture of Y atom for Fe in the minority valence band
and a reduction of the magnetic moment at Y site. The Fe
atom has four Cr and four Al as nearest neighbors and
shows a three-peak structure in both spin directions.
The exchange splitting is also weak in Fe d states and
leads to an almost symmetrical DOS in the two spin
directions. So, Fe only has a small moment of 0.04mB
antiparallel to that of Cr.
The half-metallic behavior in NiCrAl and normal ferromagnetism in FeCrAl relate to the different degree of exchange
splitting at Cr site. According to Nanda and Dasgupta [13] a
large exchange splitting in the minority spin band is preferred
in the formation of the half-metallic properties, which influence
the formation of the energy gap at the Fermi level.
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DOS (electrons / eV)
204
10
5
0
-5
-10
4
0
-4
6
3
0
-3
1.2
0.6
0.0
-0.6
Total
NiCrAl
Ni
Cr
Al(p)
0.4
0.2
0.0
-0.2
-0.4
Al(s)
-8
-6
-4
-2
0
2
4
6
Energy (eV)
Fig. 4. The total and partial DOS in ferromagnetic state for NiCrAl alloy. Minority states are on negative scale.
In order to know more about the half-metallicity in
Cr-based half-Heusler alloys, we calculated the electronic
structure of the NiCrZ (Z ¼ Al, Ga, In) alloys using
theoretical lattice constants. Their total and partial DOS
are given in Figs. 4–6, respectively. It can be seen that their
DOS structures are quite similar. As has been discussed
above, in the majority spin the Cr 3d states are basically
occupied with the high antibonding peak lying at the Fermi
level. It strongly hybridizes with Ni d states and forms a
wildly spread d band. While in the minority spin the
antibonding peak of Cr states are shifted to higher energy
above EF by the exchange splitting and cause an energy gap
at the Fermi level. So, the half-metallic properties are
obtained in the three NiCrZ half-Heusler alloys. The origin
of the gap in half-Heusler alloys NiCrZ can be attributed
to the strong hybridization between the d states of the Ni
and Cr atoms. The Fermi level lies in the bonding t2g
orbitals and antibonding eg orbitals. This is quite similar to
the formation of the gap in semiconductors such as the
well-known GaAs [23].
However, in half-Heusler alloys XYZ, Z elements
are mainly atoms with large atomic radii, like Sn, Sb or
Te. The NiCrAl or NiCrGa compound with C1b structure
is not known now due to the smaller atomic radii of
Z atoms. This instability of the bcc structure may be
explained from their electronic structures. According to the
study of Kandpal et al. [26] in CrCoSb, the high peak
in the DOS at the Fermi level may be associated with
the non-existence of CrCoSb alloy. Here, the electronic structure of NiCrZ is quite similar to that of
CrCoSb, so the high peak at EF in their DOS may have the
same effect.
In the calculated energy bands for NiCrZ, both the
maximum of the valence band and minimum of the
conduction band are at the X point so the gap is a direct
gap, which is different from normal half-Heusler alloys
with half-metallic characters in which a G–X indirect gap is
formed [23]. It is clear that the Fermi level lies above the
minority spin valence band maximum (VBM), which is the
minimum energy required to flip a minority spin electron
from the VBM to the majority spin Fermi level and is
often referred to as the ‘‘spin-flip gap (HM gap)’’ [27].
The calculated gap width around EF together with the HM
gap for the minority carriers is listed in Table 1.
The calculated total and partial magnetic moments for
the XCrAl (X ¼ Fe, Co, Ni) and NiCrZ (Z ¼ Al, Ga, In)
alloys are listed in Table 1. In half-Heusler alloys with halfmetallic character, their magnetic moment can be predicted
by the Slater–Pauling rule that is MH ¼ NV18, where MH
is the total spin magnetic moment per formula unit and NV
is the total number of valence electrons. This rule comes
from that way: in half-metals the Fermi energy is pinned in
an energy gap in only one spin direction, which leads to the
number of occupied states being an integer. So, the
Slater–Pauling rule for one atom will be expressed as
mHMF ¼ nV6 (nV is the mean number of valence electrons
per atom) for the spin magnetic moment per atom. In the
case of three atoms per unit cell, as in half-Heusler
compounds, one should subtract 18 from the total number
of valence electrons NV to get the spin magnetic moment M
per unit cell: MH ¼ NV18 [28]. It can be seen that both
the magnetic moments of CoCrAl and NiCrZ follow the
Slater–Pauling rule. They have 18 and 19 valence electrons
and magnetic moments of 0mB, 1mB, respectively. While
FeCrAl is an exception, according to the Slater–Pauling
rule, FeCrAl with 17 valence electrons should have a
moment of 1mB. However, it is not a half-metal and has a
magnetic moment of only 0.25mB.
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DOS (electrons / eV)
10
5
0
-5
NiCrGa
Total
8
4
0
-4
-8
4
0
-4
1.0
-8
0.5
0.0
-0.5
-1.0
0.5
0.0
-0.5
-1.0
205
Ni
Cr
Ga (p)
Ga (s)
-8
-6
-4
-2
0
2
4
6
Energy (eV)
DOS (electrons / eV)
Fig. 5. The total and partial DOS in ferromagnetic state for NiCrGa alloy. Minority states are on negative scale.
12
6
0
-6
10
5
0
-5
-10
10
5
0
-5
-10
0.45
0.00
-0.45
1.2
0.6
0.0
-0.6
-1.2
NiCrIn
Total
Ni
Cr
In(p)
In(s)
-8
-6
-4
-2
0
Energy (eV)
2
4
6
Fig. 6. The total and partial DOS in ferromagnetic state for NiCrIn alloy. Minority states are on negative scale.
The spin polarization ratio P is calculated as the value of
(NmNk)/(Nm+Nk) and listed in Table 1, where Nm and
Nk are the majority and minority DOS at EF, respectively.
It is clear that the P values of FeCrAl and CoCrAl are low
whereas the NiCrZ alloys predicted to be half-metal all
have a 100% spin polarization at the Fermi level.
4. Conclusion
We have investigated the electronic structures and
magnetic properties of the half-Heusler alloys XCrAl
(X ¼ Fe, Co, Ni) and NiCrZ (Z ¼ Al, Ga, In) to search
for new candidate half-metallic materials. Here we predict
that NiCrAl, NiCrGa and NiCrIn are possible half-metals
with an energy gap in the minority spin and a completely
spin polarization at the Fermi level. The energy gap can be
attributed to the covalent hybridization between the d
states of the Ni and Cr atoms, which leads to the formation
of bonding and antibonding bands with a gap in between.
Their total magnetic moments are 1mB per unit cell; agree
with the Slater–Pauling rule. Meanwhile, it is found that
FeCrAl is a normal ferromagnetic metal with a magnetic
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H. Luo et al. / Physica B 403 (2008) 200–206
moment of 0.25mB per unit cell and CoCrAl is a semi-metal
and non-magnetic.
Acknowledgments
This work is supported by National Natural Science
Foundation of China (Grant no. 50531010) and Natural
Science Foundation of Hebei (Grant no. E2006000063).
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