Materials Transactions, Vol. 45, No. 4 (2004) pp. 1065 to 1069
Special Issue on Frontiers of Smart Biomaterials
#2004 The Japan Institute of Metals
Theoretical Predict of Half-Metals in Co-Cr-Fe-Al Alloys*1
Shoji Ishida1 , Shingo Kawakami2; *2 and Setsuro Asano3
1
Department of Physics, Faculty of Science, of Kagoshima University, Kagoshima 890-0065, Japan
The Graduate School of Science and Engineering, Kagoshima University, Kagoshima 890-0065, Japan
3
National Center for University Entrance Examinations, Tokyo 153-8501, Japan
2
To theoretically examine the existence of half-metals in Co-Cr-Fe-Al alloys, band calculations were carried out for these alloys. Seeing
energy dispersion curves E(k) of Co2 CrAl with the Heusler structure, we find that the Fermi level intersects the E(k) curves in the majority-spin
state but is located at the energy gap in the minority-spin state, that is, it is predicted that this stoichiometric Co2 CrAl alloy is half-metallic. On
the other hand, the half-metallic properties are not observed in the electronic structures of Co2 FeAl. The composition dependence of electronic
structures of Co2 (Cr1x Fex )Al (x = 1/8–7/8) indicates that the alloys have the tendency to become half-metallic in the range of x < 5=8.
(Received October 9, 2003; Accepted January 15, 2004)
Keywords: half-metal, electronic structure, spintronics, energy gap, cobalt-chromium-iron-aluminum
1.
Introduction
Up to the present, a large number of the Heusler alloys
have been discovered and the magnetic properties have been
investigated experimentally and theoretically. The electronic
structures of the Heusler alloys have been calculated and the
magnetic properties have been explained on the basis of the
electronic structures. Typical ferromagnetic Heusler alloys
contain Mn atoms (X2 MnZ) where the magnetic moment is
mainly carried by the Mn atoms. Most of X2 MnZ are
ferromagnets but a few alloys exhibit antiferromagnetism.
De Groot et al. calculated electronic structures of
NiMnSb1) and PtMnSb,2,3) with half-Heusler (C1b ) structure
and showed that they are half-metals. The very high
magnetic-optical Kerr effect was observed in PtMnSb.
Schwarz4) also predicted from the electronic structures of
CrO2 that the compound is a half-metallic ferromagnet.
Hwang et al.5) observed colossal magneto resistance for the
film where grains of CrO2 are distributed in the insulating
film of Cr2 O3 .
These properties described above are beneficial for wide
technological applications related to the new field of
‘spintronics’. The half-metal is the most suitable material
in spintronics, because the electrical current is perfectly spin
polarized in the material. The half-metal enables us to
contrive new devices, controlled by magnetic field, which are
quite different from conventional devices controlled by
electrical charge. However, it has been difficult to really
synthesize half-metallic materials, so it is expected to find out
half-metals that are easily synthesized.
It was theoretically predicted that there are several halfmetals among Heusler alloys with the X2 YZ composition and
half-Heusler alloys with the XYZ composition.6–8) Recently,
Inomata et al.9) reported that Co2 Cr0:6 Fe0:4 Al alloy film
exhibited half-metallic properties. That is, they showed that
there are half-metals among Heusler alloys with a 3dtransition element on the Y site except for Mn. This result
*1This Paper was Presented at the Autumn Meeting of the Japan Institute of
Metals, held in Hokkaido, on October 11, 2003
Student, Kagoshima University
*2Graduate
motivates us to theoretically predict the existence of halfmetals in Co2 Cr1x Fex Al alloy system. We calculated the
electronic structures of Co2 Cr1x Fex Al for the various
compositions. On the basis of results, we examine the
composition and lattice parameter dependence of appearance
of half-metallicity.
2.
Crystal Structure and Method of Calculation
The L21 structure has the stoichiometric composition
X2 YZ. Because there is only one Y atom in the unit cell of
X2 YZ, the unit cell cannot be used as a unit cell of
Co2 Cr1x Fex Al where Cr atoms are partially replaced by Fe
atoms. To allow the replacement of parts of Y atoms, the L21
structure is treated as a trigonal structure (R3m, C5 3v ) with
the 160th group symmetry of ITC (International Table for
Crystallography).10) In the trigonal structure, the c-axis is
parallel to the (111) direction of the L21 p
structure
and
ffiffiffi
pffiffithe
ffi
lattice constants a and c are assumed to be 2a0 and 2 3a0
where a0 is the lattice constant of the L21 structure.
In the large unit cell of the trigonal structure, there are four
kinds of atomic sites for each constituent atoms of Co2 CrAl
and the ratio of weight of their sites is 1:1:3:3. Therefore, we
can replace one of eight Cr atoms by an Fe atom and the
system Co2 Cr1x Fex Al with the concentration of x = n/8 (n
= 0–8) is adopted as a model to examine the composition
dependence.
3.
Total Energy and Lattice Constant
To theoretically determine the lattice constant, the band
calculations were performed for several values of lattice
constants and the total energies are obtained for
Co2 Cr1x Fex Al (x = n/8, n = 0–8) as a function of the
lattice constant. As an example, we show the case of x ¼ 0 in
Fig. 1. The minimum value of the total energy gives the
theoretical value (amin ) of the lattice constant (a). The
theoretical values are summarized in Table 1. Their values
are 1–2% smaller than their experimental values11) of a ¼
0:580 nm for Co2 CrAl and a ¼ 0:578 nm for Co2 Cr0:6 Fe0:4 Al. In the following, we will discuss on the basis of
1066
S. Ishida, S. Kawakami and S. Asano
Total energy
Co2CrAl
Ferro.
Para.
0.1
0.04 aJ
Energy, E /aJ
Energy, E /aJ
Co2CrAl
Up-spin
0.05
EF
0
-0.05
-0.1
0.1
0.54
0.55
0.56
0.57
0.58
0.59
0.6
Lattice Constant, a /nm
Fig. 1 The lattice constant dependence of total energy per unit cell of
Co2 Cr1x Fex Al (x ¼ 1=2). The solid and broken lines indicate the
ferromagnetic and paramagnetic states, respectively.
Energy, E /aJ
0.53
0.05
EF
Down-spin
0
-0.05
-0.1
values
of
the
lattice
constant
Co2 Cr1x Fex Al
a/nm
x¼0
0.5687
x ¼ 1=8
0.5682
x ¼ 2=8
0.5679
x ¼ 3=8
0.5674
x ¼ 4=8
0.5673
x ¼ 5=8
0.5670
x ¼ 6=8
0.5659
x ¼ 7=8
x¼1
0.5663
0.5660
(a)
of
Fig. 2 The E(k) curves of Co2 CrAl for both spin states. The horizontal line
shows the Fermi level.
Co2FeAl
0.1
Energy, E /aJ
Table 1 The theoretical
Co2 Cr1x Fex Al.
0.05
EF
Down-spin
0
-0.05
-0.1
the electronic structures calculated at the theoretical and the
experimental lattice constant.
As an example, the total energy of the ferromagnetic state
is compared with that of the paramagnetic state for Co2 CrAl
in Fig. 1. We can confirm that the ferromagnetic state is more
stable than the paramagnetic one where half-metallicity
never appears.
4.
Energy Dispersion Curves
At first, we pay attention to the E(k) curve of Co2 CrAl,
which is shown for both spin states in Fig. 2. The Fermi level
shown by a broken horizontal line intersects the E(k) curves
in the majority (up)-spin sate but is located at the energy gap
in the minority (down)-spin state. This suggests that Co2 CrAl
is a half-metal where electrons at the Fermi level are 100%
spin polarized. That is, it is expected that the alloy shows
typical metallic behavior for the up-spin states while at the
same time being a semiconductor for the down-spin states.
The Fermi level intersects the E(k) curves in the up-spin
states for all of alloys. Therefore, we pay attention to only the
down-spin states to judge whether the alloy is half-metallic or
not, in the following discussion.
Next, the E(k) curves of Co2 FeAl are shown in Fig. 3
where the Fermi level intersects the E(k) curves in the downspin state. Comparing the E(k) curves of Co2 FeAl with those
of Co2 CrAl, we notice that the energy gap is formed at
around the X-point in Co2 CrAl but not in Co2 FeAl. Then,
band structures were calculated for the Co2 Cr1x Fex Al (x =
Fig. 3 The E(k) curve of Co2 FeAl for the down-spin state. The horizontal
line shows the Fermi level.
n/8, n = 1–7) system to examine how the Fe atoms affect the
formation of energy gap. Their E(k) curves are shown in
Fig. 4 for various concentration x of Co2 Cr1x Fex Al. The Xpoint of the L21 structure in Figs. 2 and 3 corresponds to the
-point of the trigonal structure. The Fermi level is
positioned at the gap for the case of x ¼ 1=8 and 2/8 but
intersects the E(k) curves for x > 3=8. The curves intersecting the Fermi level for x > 3=8 affect the half-metallicity and
formation of the energy gap. To determine the character of
these curves, the composition of the eigenfunctions at the Xpoint was analyzed. The composition is 96.34% d-state of Cr
for Co2 CrAl, 12% d-state of Cr and 84.85% d-state of Fe for
Co2 Cr1=2 Fe1=2 Al, and 97.18% d-state of Fe for Co2 FeAl. This
means that the d-composition of Fe increases with increasing
concentration (x) of Fe through the hybridization between dstates of Cr and Fe, and Fe atoms play an important role in the
half-metallicity and formation of the energy gap.
Furthermore, the lattice constant (a) dependence of the
half-metallicity was examined. The E(k) curves of x ¼ 3=8
are shown in Fig. 5 for three values of the lattice constants.
The theoretical lattice constant is amin ¼ 0:5673 nm which
was determined by the lattice constant dependence of the
total energy as shown in Fig. 1. It is found that
Co2 Cr5=8 Fe3=8 Al is not half-metallic at amin but half-metallic
for a > 0:57 nm, and the Fermi level is situated at the bottom
Theoretical Predict of Half-Metals in Co-Cr-Fe-Al Alloys
Energy, E /aJ
Co2Cr1-xFexAl
0.05
x=1/8
EF
Energy, E /aJ
0
Co2Cr5/8Fe3/8Al (Down-spin)
0.15
-0.05
0.1
0.05
0
x=2/8
EF
0.05
-0.05
0.1
0.05
0
0
Energy, E /aJ
-0.05
0.1
0.05
x=4/8
EF
a =0.56734nm
EF
a =0.57nm
-0.15
0.15
x=3/8
EF
EF
-0.05
0.05
-0.05
-0.05
0.1
0.05
0
-0.15
0.15
x=5/8
EF
-0.05
0.1
0.05
Energy, E /aJ
Energy, E /aJ
Energy, E /aJ
Energy, E /aJ
Energy, E /aJ
Energy, E /aJ
Energy, E /aJ
0.1
1067
x=6/8
EF
0
-0.05
0.1
0.05
0
x=7/8
EF
0.05
-0.05
a =0.58nm
EF
-0.15
-0.05
Fig. 5 The lattice constant (a) dependence of the E(k) curves of
Co2 Cr5=8 Fe3=8 Al. The horizontal line shows the Fermi level.
Fig. 4 The E(k) curves of Co2 Cr1x Fex Al for several values of concentration x. The horizontal line shows the Fermi level.
0.2
Co2FexCr1-xAl
0.15
0.08
x=0
x=2/8
x=4/8
x=1/8
x=3/8
x=5/8
x=1/8
x=3/8
x=5/8
0.1
∆n
Gap Width, W /aJ
Co2Cr1-xFexAl
x=0
x=2/8
x=4/8
0.06
0.05
0.04
0
0.02
-0.05
0.55
0.56
0.57
0.58
Lattice Constant, a /nm
0.59
0
0.55
0.56
0.57
0.58
Lattice Constant, a /nm
0.59
Fig. 7 The lattice constant (a) dependence of the decimal fraction n of
the number of down-spin electrons for Co2 Cr1x Fex Al.
Fig. 6 The lattice constant (a) dependence of the energy gap width W of
Co2 Cr1x Fex Al.
of the conduction band for a ¼ 0:57 nm. The value of
a ¼ 0:57 nm is 0.45% larger than the theoretical value of
lattice constant. Thus, at the theoretical lattice constant, the
value of x ¼ 3=8 is the critical value to judge whether
Co2 Cr1x Fex Al becomes half-metallic or not.
To discuss the half-metallicity from a different point of
view and summarize above features, we pay attention to the
energy gap width (W) and the number of down-spin electrons
per unit cell (n). When the value of n is an integer, we can
judge an alloy is a half-metal. We use the symbol ‘n’ as the
decimal fraction of the number of down-spin electrons. The
lattice constant (a) dependence of W and n are shown in
Figs. 6 and 7, respectively for Co2 Cr1x Fex Al alloy. A band
gap appears for large values of a when x < 5=8. From these
figures, it is expected that Co2 Cr1x Fex Al becomes halfmetallic when x < 3=8, and for a > 5:7 nm when 3=8 <
x < 5=8. That is, half-metallic properties may be observed
with high possibility for Co2 Cr1x Fex Al (x < 5=8).
1068
5.
S. Ishida, S. Kawakami and S. Asano
Local Density-of-State and Magnetic Moment
Alloys with half-metallicity have an integer value of spin
polarization per unit cell. In Table 2, the values of spin
polarization at amin and a ¼ 0:58 nm are summarized for the
constituent atoms and the unit cell of Co2 Cr1x Fex Al. We can
see that the value per unit cell is an integer for the halfmetallic alloys but the value per atom is not an integer. The
values of Co and Cr in these alloys are around 0.9–1.2 B and
the direction of the spin polarization of Al is antiparallel to
the others although the magnitude is small. The values 2.7–
2.9 B of Fe are fairly lager than the magnetic moment about
2.2 B of pure iron. These features will be examined on the
basis of the density of state (DOS) in the following.
The local DOS of Co and Y(Cr or Fe) in Co2 YAl are
shown in Figs. 8 and 9. The Fermi level of Co2 CrAl is
situated near the peaks of local DOS of Co and Cr in the upspin state but at the energy gap in the down-spin state. On the
other hand, the Fermi level of Co2 FeAl is situated at the tails
of high peak of local DOS of Co and Fe in the down-spin
state. These features correspond to the prediction of the half-
metallicity deduced from the E(k) curves.
Because the number of d-electorns of Fe is larger than that
of Cr, the potential around Fe is deeper than that of Cr.
Therefore, the DOS of Fe is located in lower energy range,
compared with the DOS of Cr. It is found that the differences
in the magnetic moments are reflected on the differences in
occupancy of local DOS of Co, Cr and Fe.
To examine the enhancement of magnetic moment on Fe
in Co2 FeAl, we pay attention to the neighbor atoms of Fe
atoms. The first nearest neighbor atoms of Fe in Co2 FeAl are
Co atoms while those in pure iron are Fe atoms. This suggests
that d-states of Fe in Co2 FeAl hybridize with d-states of the
first nearest neighbor atoms in lower energy range, compared
with Fe atom in pure iron, because the potential around Co is
deeper than that of Fe. Certainly, the DOS of Co and Fe of the
up-spin state have some peaks in the same energy range, that
is, the d-states of Co and Fe hybridize strongly. Therefore, the
band tail (hole) of up-spin states above the Fermi level is
small and the Fe d-states of the up-spin state are mostly
occupied. Thus, the magnetic moment of Fe atom is
enhanced in Co2 FeAl.
Table 2 The magnetic moments due to spin polarizations in Co2 Cr1x Fex Al. The values for constituent atoms and the unit cell (u.c.) are
given in unit of B for the lattice constants of amin and a ¼ 0:58 nm.
x¼0
x ¼ 1=8
x ¼ 2=8
x ¼ 3=8
x ¼ 4=8
x ¼ 5=8
x ¼ 6=8
x ¼ 7=8
Co
0.92
0.95
0.99
1.03
1.06
1.09
1.12
1.15
Cr
1.24
1.23
1.21
1.21
1.20
1.21
1.23
1.17
2.82
2.82
2.77
2.76
2.74
2.72
2.72
2.72
Al
0:08
0:09
0:09
0:09
0:10
0:10
0:11
0:11
0:11
u.c.
3
26
28
29.99
31.95
33.95
35.86
37.82
4.98
a ¼ 0:58
x¼0
x ¼ 1=8
x ¼ 2=8
x ¼ 3=8
x ¼ 4=8
x ¼ 5=8
x ¼ 6=8
x ¼ 7=8
x¼1
Co
0.9
0.92
0.97
1.01
1.04
1.08
1.10
1.14
1.18
Cr
1.29
1.31
1.27
1.25
1.23
1.23
1.30
1.27
2.87
2.87
2.82
2.82
2.79
2.78
2.78
2.78
0:09
3
0:10
26
0:10
28
0:11
30
0:11
32
0:12
34
0:12
37.98
0:12
38.00
0:13
5.01
a ¼ amin
Fe
Fe
Al
u.c.
x¼1
1.19
25
Co d-state
20
DOS/(aJ atom spin)-1
14
12
10
8
6
4
2
0
Up-spin
Down-spin
Co2FeAl
Co d-state
16
12
Up-spin
Down-spin
8
4
0
Cr d-state
Up-spin
Down-spin
20
DOS/(aJ atom spin)-1
DOS/(aJ atom spin)-1
DOS/(aJ atom spin)-1
Co2CrAl
15
10
5
Fe d-state
Up-spin
Down-spin
25
20
15
10
5
0
0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Energy, E /aJ
Fig. 8 The local DOS of Co and d-states of Cr in Co2 CrAl. The solid and
broken lines distinguish the ferromagnetic and paramagnetic states,
respectively. The vertical line indicates the Fermi level.
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Energy, E /aJ
Fig. 9 The local DOS of Co and d-states of Fe in Co2 FeAl. The solid and
broken lines distinguish the ferromagnetic and paramagnetic states,
respectively. The vertical line indicates the Fermi level.
Theoretical Predict of Half-Metals in Co-Cr-Fe-Al Alloys
6.
Summary
We paid our attention to Co2 Cr1x Fex Al alloys and
calculated the band structures to theoretically examine the
existence of half-metals in these alloys. The results suggest
that the Co2 Cr1x Fex Al alloys have the tendency to become
half-metallic in the concentration range of x < 5=8. We also
found that the magnetic moment on Fe atom is enhanced in
Co2 Cr1x Fex Al, compared with that in pure iron. This is
because the d-states of Fe in the majority-spin state are
mostly occupied by electrons, owing to the hybridization
between the d-states of Fe and those of Co.
Acknowledgment
The authors are grateful to Professor Koichiro Inomata at
Tohoku University who gave us significant imformation.
1069
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