Materials Transactions, Vol. 47, No. 3 (2006) pp. 599 to 606
Special Issue on Shape Memory Alloys and Their Applications
#2006 The Japan Institute of Metals
The Electronic Structure and Magnetic Properties
of Full- and Half-Heusler Alloys
Svetlana E. Kulkova1 , Sergey V. Eremeev1 , Tomoyuki Kakeshita2 ,
Sergey S. Kulkov1 and Gennadiy E. Rudenski1
1
Institute of Strength Physics and Materials Science of the Russian Academy of Sciences,
Siberian Branch, Tomsk, 634021, Russia
2
Department of Materials Science and Engineering, Graduate School of Engineering, Osaka University,
Suita 565-0871, Japan
The electronic structure of the full- and half-Heusler alloys have been studied by ab-initio calculations using full potential augmented
plane-wave-method (FLAPW). It was shown that obtained equilibrium lattice parameters and magnetic moments agree well with available
experimental data. The influence of vacancies on the electronic structure and magnetic properties of Ni2x MnGa and Co2x ZrSn is analyzed.
(Received August 22, 2005; Accepted December 15, 2005; Published March 15, 2006)
Keywords: Heusler alloys, electronic band structure, magnetic properties
1.
Introduction
Ferromagnetic Heusler alloys exhibited magnetic shape
memory effect, magnetic field induced superelasticity and
large strain-induced changes in the magnetization are
recently very attractive subject of current researches.1–17)
The ferromagnetic martensites are experimentally found in
different Heusler alloys such as Ni–Mn–Ga, Ni–Fe–Ga, Co–
Ni–Ga and other alloys. The magnetic shape memory effect
is related to martensite transformations that are sensitive to
pronounced magnetoelastic interaction. A high efficiency of
the magneto-mechanical properties makes these materials
very attractive for applications as different kind of actuators,
sensors, magnetic micro-electro-mechanical systems, for the
recording and storage of information, etc. The current
advantages in new materials are promising for engineering
of new spintronic devices. In this context the problem of local
magnetic properties can be one of the most important in the
physics of these materials.
It is known that magnetic properties of Heusler alloys are
strongly dependent on both the conduction electron concentration and chemical bonding. For example Mn-based
compounds demonstrate rather localized magnetism due to
configuration of Mn d-orbitals,9,13,17) whereas Co-based
compounds show more itinerant behavior.18,19) So-called
full-Heusler alloys with a general formula unit of X2 YZ,
where X and Y denote the transition metals and Z is s-p
element such as Al, Ga, Sn, Sb, etc., are studied also with
respect to the transition from the ferromagnetic phase to an
antiferromagntic one with changing of the concentration of
the carriers. Half-metallic Heusler ferromagnets (XYZ) have
an energy gap for minority spin bands and the conduction
electrons at the Fermi level (EF ) show 100% spin polarization
and can be used as spin-polarized electron sources along with
metal oxides and III–V group semiconductors. It is known
that the full-Heusler alloys such as Co2 MnZ with Z ¼ Si, Ge
demonstrate half-metallic behavior also.
The magnetic properties of the Heusler alloys are very
sensitive to the local geometry and chemical composition. In
order to understand the dependence of magnetic properties on
the atomic composition and crystal structure the ab-initio
investigation of the electronic structure can be very useful.
The Ni2 MnGa remains most investigated Heusler alloy by
both experimental and theoretical approaches. It is known
from the literature that the physical properties of this alloy
very sensitive to structural disorder and deviations from
stoichiometric composition. At present exist several band
structure calculations of Ni2 MnGa which were performed
using different ab-initio techniques, in particular augmentedspherical-wave method,9) full-potential linearized augmented-plane-wave method,12,17) pseudopotential plane-wave
approach,20) etc. Less attention was paid to other alloys but
a number of the theoretical investigations of the full-Heusler
alloys are sharply increased.21–27) The similar tendency is
observed for half-metallic Heusler alloys.28–33) The calculations were mainly performed using the local spin density
approximation (LSDA), which are known to be underestimated lattice constants and provide smaller magnetic moments. In present paper we report the results of ab-initio
calculations of the electronic structure and structural and
magnetic properties in different series of Heusler alloys based
on Fe-, Co- and Mn, focusing on the effects of their
composition and s-p Z atom and extend our study on the halfHeusler alloys also using the full-potential approach within
the generalized gradient approximation (GGA) for exchange
correlation potential.
2.
Computational Details
The FLAPW method (Wien implementation34)) within
GGA according to the Perdew–Burke–Ernzerhof parameterization35) was applied to the band structure calculations of
alloys. The core states were treated fully relativistically,
whereas semicore states (3s-, 3p- for Mn, Fe, Co, Cu), which
were treated as the valence electrons and the valence states
were calculated semi-relativistically. The GGA including
nonlocal contribution gives more accurately description of
the energy bands. The multipole expansion of the crystal
potential and electron density within the atomic spheres was
cut at l ¼ 12. Nonspherical contribution to the charge density
600
S. E. Kulkova, S. V. Eremeev, T. Kakeshita, S. S. Kulkov and G. E. Rudenski
Table 1
Structural and magnetic properties of some Heusler alloys, the values in brackets are experimental ones.
a (nm)
tot (B )
X
Y
Z
Ni2 MnAl
0.5791 (0.5822a )
4.00
0.35
3.34
0:03
Ni2 MnGa
0.5813 (0.5825a )
4.05 (4.17a )
0.35
3.35
0:05
4.09b
0.37b
3.36b
0:04b
3.98 (4.22a )
0:07
b
0.5812
0.6064 (0.6053a )
Ni2 MnSn
0.23
3.57
Co2 MnAl
0.5694 (0.5756 )
4.10 (4.01a )
0.73
2.69
0:04
Co2 MnGa
0.5756c
0.5728 (0.5770a )
3.97c
4.09 (4.05a )
0.77c
0.76
2.53c
2.75
0:10c
0:08
0.5770c
4.06c
0.69c
2.78c
0:09c
a
a
Co2 MnSn
a
0.5984 (0.6000 )
4.99 (5.08 )
0.95
3.19
0:09
0.6000c
4.98c
0.93c
3.20c
0:08c
5.0d
0.99d
3.09d
0:05d
0.5940 (0.5949 )
3.47
0.05
3.40
0:07
0.5949e
3.53e (3.6–4.12e )
0.07e
3.49e
0:05e
Ni2 CoGa
0.5725 (0.5725a )
0.5720
1.71
1.78b
0.13
0.16b
1.55
1.55b
0:02
0:02b
Fe2 CoGa
0.5781 (0.5767a )
6.14, 5.49LSDA (5.09a )
2.26
1.85
0:07
0.5779
6.05b
2.20b
1.83b
0:07b
Fe2 NiGa
0.5781 (0.5780a )
4.71
2.20
0.56
0:11
Co2 FeAl
0.5725 (0.5730a )
5.0
1.12
2.71
0:10
0.5730c
4.89c
1.13c
2.73c
0:10c
0.5964d
a
Cu2 MnAl
b
Co2 FeSi
0.5660 (0.5658a )
5.59
1.41
2.84
0.01
Co2 FeGa
0.5658c
0.5714 (0.5737a )
5.27c
5.0
1.27c
1.21
2.76c
2.79
0:03c
0:07
Co2 VAl
0.5738 (0.5770a )
1.99 (1.65a )
0.94
0.22
0:03
1.93c
0.86c
0.23c
0:03c
2.00 (1.95a )
0.98
0.15
1.01 (071a )
0.60
0:06
0.13c
0.07c
0:0c
f
f
c
0.5770
0.5754 (0.5786a )
Co2 VGa
a
Co2 TiAl
0.5825 (0.5848 )
c
0.5848
Co2 TiGa
Co2 TiSn
0.82
a
b
c
0:05
0:0c
f
0:03f
0.99 (0.75 )
2.00 (1.93a )
0.63
1.05
0:03
0:09
0.6073c
1.78c
0.91c
0:04c
0:0c
f
f
f
0:0f
d
e
0.92
0:02
0:01
0:0
f
Refs. 2, 3); Ref. 12); Ref. 25); Ref. 23); Ref. 8); Ref. 19).
and potential within spheres were considered up to lmax ¼ 6.
The plane-wave cutoff of RKmax ¼ 9 leads to about 400
basis function. In the interstitial region the charge density and
the potential were expanded as a Fourier series with wave
vectors up to Gmax ¼ 0:74 nm (14 a.u.1 ). In FLAPW
methods no shape approximation to the potential occurs
and it is among the most accurate band structure methods at
present. A convergence test of the total energy with respect to
the number of k-points in an irreducible part of Brillouin zone
indicates that 47–72 k-points is sufficient to get the energy
precision of 2:18 1021 J (1 mRy). The unit cell of most
full-Heusler alloys is a face centered cubic cell with basis
atoms in (1/4, 1/4, 1/4) and (3/4, 3/4, 3/4) positions for X,
(1/2, 1/2, 1/2) for Y, and (0, 0, 0) for Z. The half-Heusler
alloys correspond to XYZ composition (C1b structure) where
one site for X atom is empty.
3.
0.42
0:02
0:0
0.5831 (0.5848)
0.6085 (0.6073)
1.82
a
i (B )
X2 YZ
Electronic Structure of Full-Heusler Alloys
Electronic structure of several advanced ferromagnetic
full-Heusler alloys was calculated at the first step for
experimental lattice parameters. After that the volume
optimization using the calculations for several lattice
parameters near experimental ones were performed. The
equilibrium lattice constants and magnetic moments are
presented in Table 1. It is seen that theoretical lattice
constants agree well with experimental data.2,3) The maximal
error for lattice parameters is less than 1.0%. The obtained
values of the total and local magnetic moments are also in
good agreement with available experimental data2) and
previous ab-initio calculations.8,12,19,23,25) The results for
two series of alloys Ni2 MnZ and Co2 MnZ with Z ¼ Al, Ga,
Sn show that the Mn magnetic moment depend on the Z
element but the effect is less expressed for the first (Ni2 MnZ)
series. The magnetic moment of Z element is negative and it
is not large as 0:1 B . The decrease of Mn free-atom
moment is attributed to strong hybridization of Mn delectrons with Al (Ga,Sn) p-electrons in Ni2 MnZ alloys. The
minority-spin electrons of Mn are not occupied in these
alloys as in Cu2 MnAl alloy.8,13) We would like to point out
N(E) /(states/eV/cell)
The Electronic Structure and Magnetic Properties of Full- and Half-Heusler Alloys
5
Co2MnAl
EF
0
-5
5
Co2MnGa
0
-5
5
Co2MnSn
0
-5
-8
Fig. 1
-4
0
Energy, E / eV
4
Spin-resolved DOS for Co2 MnZ alloys with Z is Al, Ga and Sn.
that the increase in the lattice constant caused by the larger Sn
atom could be a reason of small change of Ni and Mn
magnetic moment in the case of Ni2 MnSn alloy. As
mentioned in Ref. 25) many full-Heusler alloys can follow
the Slater–Pauling (SP) curve. The total moment obeys the
rule M = Z-24, where Z is total number of the valence
electrons and 24 means that there are 12 occupied spin-down
states per unit cell. Thus the Ni2 MnSn alloy with 29 valence
electrons should have the total magnetic moment of 5 B that
does not agree with experimentally obtained one. Actually in
Ni2 MnZ and Cu2 MnZ alloys the magnetic moment can not
be substantial larger than magnetic moment on Mn atoms
because only Mn determined the magnetic behavior of these
alloys. Note that both majority (up) and minority (down) spin
band structures of Ni2 MnZ are strongly metallic. It is known
that in Co2 MnZ series if Z is Si or Ge the minority states are
found by experimental and theoretical studies to be semiconducting23) but other alloys of this series have spin-down
gaps with small densities of states at the Fermi level.25) Our
results are in good agreement with those obtained for Z ¼
Sn23) and Z ¼ Al, Ga, Sn.25) In the case of Co2 MnZ with
Z ¼ Al, Ga and Sn the minority spin states are slightly above
the Fermi level (Fig. 1). The calculated magnetic moment for
Co2 MnSn is not integer as in Ref. 23) but it is slightly higher
than that in Ref. 25). Note that the total value of spin
magnetic moment is found to be 5.04 B within TB-LMTOASA LSDA calculation in Ref. 27). As mentioned in
Ref. 25) the Mn magnetic moment increases only slightly
along Si-Ge-Sn isoelectronic series but it increases much
more in Al–Ga–Sn series in quite good agreement with
results of Ref. 27). This effect can be ascribed as increase of
the exchange interaction due to larger lattice constant for
Co2 MnSn (the change of lattice parameter is 0:03 nm) that
lead to smaller overlapping of d-states on the Mn sublattice.
The magnetic moment in the series Co2 MnZ is in the line
with the rule of M = Z-24. The number of valence electrons
increases by one in Co2 MnSn compared with Co2 MnAl(Ga)
alloys that results to tot ¼ 4:99 B . The replacement of Ni
by Co leads to decrease of Mn magnetic moment down to
601
2.69–2.75 B in Co2 MnAl(Ga) but the total magnetic moment changes insignificantly due to Co contribution. The Co
magnetic moment increases along Al–Ga–Sn series because
decreasing of Co–Mn hybridization for minority spin states.
In second considered family of alloys Co contributes in the
total magnetic moment twice more than Ni atoms. It is
interesting to point out that the value of Co spin moment
remains substantial less in the alloys in comparison with the
pure metal. A drastic decrease of magnetic moment down to
1.71 B occurs if Mn atom in the Ni2 MnGa is replaced by Co
atom. Actually the formation of magnetic properties is more
complicated when alloy has two magnetic elements in its
composition because different mechanisms should be taken
into account including superexchange through s-p electrons
of Z element, the interaction between the s-conduction
electrons and Mn localized d-electrons and the Mn-X
interaction where X is transition metal.
Among considered Heusler alloys Fe2 CoGa has a largest
total magnetic moment (6.14 B ). This value agrees well
with earlier calculation (6.05 B )12) but contradict the Z-24
rule. As mentioned above the alloy with 28 valence electrons
must have the total magnetic moment of 4 B that does not
agree with experimentally obtained one of 5.09 B .2,3) It
should be noted that the estimation of the magnetic moment
within LSDA gives the value of 5.49 B . The new
experimental investigation would be desirable for this alloy.
The decrease of tot is observed if Co atom at Y site is
replaced by Ni in Fe2 YZ family of alloys due to small
contribution of Ni in the spin magnetic moment. It is known
that in most Heusler alloys the atoms in the Y sites have a
large magnetic moment while the moments at X site are
usually small except few cases including X ¼ Fe mentioned
above. Other samples are Co2 FeZ and Co2 MnZ alloys. The
spin densities of states in the case of several Co2 YZ alloys are
shown in the Fig. 2. There is a strong splitting of DOS for
majority and minority electrons that leads to appearance of
significant spin magnetic moment in Co2 FeZ alloys whereas
the splitting of spin subbands is substantial less in alloys with
V or Ti in the Y site. This change in the magnetic behavior is
due to smaller Co–Ti (V) hybridization in comparison with
Co–Fe interaction. Depending of the number of valence
electrons of Z element (3 for Al and Ga or 4 for Si and Sn)
Co2 FeZ or Co2 MnZ alloys have 28–30 valence electrons and
can reach large spin moments (Table 1). The alloys with Ti
or V at the Y site have the moment of 1–2 B only. We found
these systems to be weakly itinerant with magnetic moments
in the satisfactory comparison with experiment as in
Refs. 19, 25). In the series Co2 TiZ the Ti local moment is
negative and coupling between Co and Ti is antiparallel. The
Co local magnetic moment is not consistent in this series of
alloys (Table 1) and it is larger in Co2 TiSn due to increase of
valence electrons in Sn in comparison with Al and Ga as well
as the lattice expansion in this case. In this family of alloys
the total spin magnetic moment is determined by Co atoms
that are consistent with experimental results.36) Our results
for Co2 TiAl and Co2 TiSn are slightly different from those in
Ref. 19) due to different approximation used in calculations
but the total moment is substantial higher than that obtained
in Ref. 25) by full-potential screened Korringa–Kohn–
Rostoker (FSKKR) method. It is equal to 0.13 B for
15
Co2FeAl
10
Co2FeGa
5
Co2FeSi
EF up
0
-5
-10
-12
Magnetic moment, M / µ B
Fig. 2
down
-8
-4
0
4
Energy, E / eV
up
Co2VGa
5
0
-5
down
-10
-12
-8
-4
0
Energy, E / eV
4
Spin densities of states of several Co-based Heusler alloys calculated by FLAPW methods.
total
Fe
3.6
3.2
Co
Ga
1.5
1.0
2.8
-0.1
2.4
0.8
Fig. 3
EF
Co2VAl
10
Magnetic moment, M / µ B
-15
N(E) /(states/eV/spin)
S. E. Kulkova, S. V. Eremeev, T. Kakeshita, S. S. Kulkov and G. E. Rudenski
N(E) /(statest/eV/spin)
602
1.0
1.2
c /a
1.4
1.6
0.8
1.0
1.2 1.4
c /a
1.6
Total magnetic moment and contributions associated with individual atoms in Fe2 CoGa as a function of c=a.
Co2 TiAl but it is 1.78 B for Co2 TiSn.25) The last value
agrees satisfactory with ours result (Table 1). Actually the
spin magnetic moments in this series are overestimated
within GGA by 0.2–0.3 B . Both Co2 VAl and Co2 VSn
alloys have the total magnetic moment of 2 B and they
follow the SP curve. So, the magnetic behavior of ferromagnetic Heusler alloys can be controlled in particular by the
Z atom through its number of the valence electrons that can
be used for practical application.
Since the primarily role in martensitic transformations in
Heusler alloys belongs to the lattice distortions we estimated
c=a dependence of magnetic moment for Ni2 MnGa and
Fe2 CoGa alloys. As it was shown in previous investigation
within pseudopotential approach for Ni2 MnZ with Z ¼ Al,
Ga, In Ref. 20), the total magnetic moments in these alloys
showed the sharp minimum at c=a 1 and local maximum at
c=a 0:94. This feature was attributed to Ni contribution,
which follows the total magnetic moment variation in this
region. The same trend was obtained in the present
calculation. It was shown the Mn magnetic moment changes
slowly at the range c=a ¼ 0:9{1:15 and has several local
minima but it sharply decreases at c=a > 1:2 in consistent
with result of Ref. 20). As seen from Fig. 3, presented the
atomic decomposition of total magnetic moment vs. c=a for
Fe2 CoGa, the maximum of the curve near c=a 1 is
determined by all components of alloy but iron has only
local maximum in this region. Contrary to Ni2 MnGa, where
X element determines the behavior of the c=a behavior of the
total curve in Fe2 CoGa the Co contribution is responsible for
this effect. As seen from consideration of magnetic properties
of Heusler alloys the local magnetic moment on both Ni and
Co atoms are very sensitive to change of the lattice geometry.
So, the lattice distortions provide us the way in increasing the
magnetic moment.
Finally we studied the change of the electronic structure of
Ni2x MnGa alloys with lower Ni concentration. In this case
one of two Ni sites in the L21 unit cell remains fully occupied
whereas the other site can be replaced by vacancies. The
reduction of Co concentration in Co2 TiSn and Co2 ZrSn
alloys in such approach demonstrated the itinerant magnetic
magnetism.21) Obtained results for Co2x ZrSn systems are in
the line with LMTO calculations.21) In this series of alloys Co
determines the value of total spin moment. We obtained
1.08 B for Co (0.94 B 21)). As mentioned above the
magnetic moment at the transition metals site can be
overestimated within GGA compared to LSDA. The total
moment is of 0.2 B higher than the experimental value
1.81 B .37) The equilibrium lattice constant is found to be
0.6290 nm (experimental value is 0.6249 nm3)). The appearance of vacancies leads to decrease of the Co magnetic
moment on the vacancy sublattice down to 0.61 B for x ¼
0:25 (0.58 B 37)). The average Co magnetic moment is
0.52 B for x ¼ 0:5 (0.46 B 21)). We save L21 structure for
alloys with decrease of Co concentrations that is in consistent
N(E)/(states/eV/cell)
The Electronic Structure and Magnetic Properties of Full- and Half-Heusler Alloys
20
Ni 2 MnGa
EF
0
-20
20
Ni 1.75MnGa
0
-20
20
Ni 1.5 MnGa
0
-20
-8
Fig. 4
-4
0
Energy, E/eV
4
The spin DOS of Ni2x MnGa as a function of Ni concentration.
with experimental data. The electronic structure results
showed pseudogap formation below the Fermi level but this
alloy is not classified as a half-metallic ferromagnet. Note
that more substantial change in Co magnetic moment was
found on sublattice with Co deficiency. The densities of
states for Ni2 MnGa with Ni deficiency are shown in the
Fig. 4. The changes are connected primarily with Ni spin
subbands. The shape of DOS below the Fermi level for spindown is determined mainly by Ni d-states. As seen from the
Fig. 4, Ni deficiency influences slightly the DOS near EF and
leads to the narrowing of Ni-subbands. This results on
delocalization of Mn states. Our estimation showed that the
Mn magnetic moment is varied from 3.35 up to 3.38 B and
the Ni magnetic moment decreases within 0.05 B at the
sites near the Ni-vacancy for x ¼ 0:25. In this case the
contraction of the lattice constant is occurred
(ax¼0:25 ¼ 0:5761 nm). The following reduction of the Ni
concentration (x ¼ 0:5) increases the local moment at the Mn
site up to 3.39 B . In this case the average Ni magnetic
moment is 0.29 B that is comparable with its change due to
lattice distortions. The effects of disordering on the transition
metals sublattice can influence on the temperature of
martensitic transformation. The studying of the role of
structural defects is very important in this context. This work
is in progress now and will be reported in our forthcoming
paper.
4.
Electronic Structure of Half-Heusler Alloys
The electronic structure of the half-metallic Heusler alloys
was studied last decade by experimental and theoretical
approaches28–33) but basically the linearized muffin-tin orbital
method was used with limitation for shape approximation for
crystal potential excepting paper Ref. 30), where the fullpotential screened Korringa–Kohn–Rostoker method was
applied. The full-potential approach is more desirable for
better description of magnetic properties in the structures
with possible aspherical spin density distribution. The
603
position of the Fermi level with respect to the band gap can
be also quite sensitive to the method of the band structure
calculation. We have performed the study of equilibrium
lattice constant and calculated magnetic moments in the
family of XMnSb alloys with different transition metals at the
X site. It should be noted that the experimental lattice
parameters were used in Ref. 30) for more precise description of magnetic properties because the authors were applied
LSDA that underestimated the lattice parameters. This can be
a reason of quite good agreement between results obtained in
Ref. 30) and in present study. Actually LSDA and GGA
gives the same trends with respect to energy gap and position
of the Fermi level but the difference in the magnetic moments
can be presented. Obtained results compared with experimental data and those from Ref. 30) are presented in
Table 2. As for the full-Heusler alloys the equilibrium
constants obtained from the ferromagnetic calculations for
half-Heusler alloys agree well with experimental data. The
calculated band structure for NiMnSb and PtMnSb are in
good comparison with previous calculation.28) Spin polarized
DOS for several alloys are shown in the Fig. 5. The lowest
band located basically in the region 19:22 1019 –
16:02 1019 J (12–10 eV) below the Fermi level for
considered alloys have mainly Sb 5s character. The Fermi
level cuts the majority spin-up bands, which are metallic and
lies in the gap of minority spin-down band in the case of
X ¼ Ni and Pt that reflect their semiconducting behavior. In
the case of NiMnSb the calculated indirect -X energy gap
equals to 0:88 1019 J (0.55 eV) that is slightly less than the
experimental value 1:12 1019 J (0.70 eV) but it is in good
comparison with result of 0:88 1019 J (0.55)28) and
0:80 1019 J (0.5 eV).30) The position of main Mn and Ni
peaks for both spin direction agrees also well with corresponding theoretical data.28,30) It is well known that Mn
minority bands are fully unoccupied but Ni bands are fully
occupied for both spin directions in NiMnSb. The magnetic
moment at Mn atom is 3.68 B (3.76 B 28) and 3.71 B 30))
whereas Ni has small local moment 0.27 B (0.25 B 28) and
0.26 B 30)). The contribution of Sb is negative (0:07 B ).
In the case of the PtMnSb the obtained energy gap is
1:38 1019 J (0.86 eV) that agrees also quite well with the
experimental value of 1:44 1019 J (0.90 eV).38) In this
case the minority bands lay only 0:06 1019 J (0.04 eV)
above the Fermi level if the spin–orbital interaction is not
taken into account. The calculated value of the magnetic
moment (3.84 B ) at the Mn site is slightly less than in
Refs. 28, 30), but it is increased if Pt is replaced by Pd or Cu
(Table 2). Ni, Pd, Pt, Cu, Au are ferromagnetically coupled
to Mn but in the case of XMnSb compounds with Co, Rh and
Ir their spin magnetic moments are antiparallel to the Mn
moment. Actually all considered Mn-based alloys have the
large total magnetic moment due to the Mn contribution as in
full-Heusler alloys. The reduction of the total magnetic
moment in the alloys with X metals from Co-row (Co, Rh, Ir)
is partly due to negative contributions of both X and Sb atoms
and it is decreased by the increasing hybridization in the
minority band between the Co and Mn d-states. In the case of
CoMnSb the Fermi level is located a little below the left edge
of the gap. It is much wider (1:44 1019 J (0.9 eV)) than
that in NiMnSb alloy. The origin of the gap was discussed in
604
S. E. Kulkova, S. V. Eremeev, T. Kakeshita, S. S. Kulkov and G. E. Rudenski
Table 2 Total and partial magnetic moments and DOS at the Fermi level for half-Heusler XMnZ alloys, the experimental data3Þ are given
in brackets.
Alloys
atheor (aexp ) nm
i (B )
tot (B )
Me
NiMnSb
PdMnSb
PtMnSb
CoMnSb
RhMnSb
IrMnSb
a
0.5909 (0.5920)
Mn
Sb
4.00 (3.85)
0.27
3.68
0:06
3.96a
0.26a
3.71a
0:06a
0.6214 (0.6246)
4.07 (3.95)
0.10
3.94
0:09
0.6224 (0.621)
4.02a
4.00 (4.14)
0.08a
0.12
4.01a
3.84
0:11a
0:07
3.94a
0.09a
3.89a
0:08a
0.5818 (0.5853)
0.6095 (0.6152)
3.00 (4.0)
0:18
3.25
0:08
2.96a
0:13a
3.18a
0:10a
3.37 (3.63)
0:10
3.60
0:11
3.29a
0:13a
3.57a
0:14a
0.5929 (0.6164)
3.11 (3.10)
0:15
3.39
0:09
CuMnSb
0.6096 (0.6095)
3.02a
4.08 (3.90)
0:19a
0.11
3.33a
3.97
0:11a
0.03
4.34a
0.13a
4.10a
0.03a
AuMnSb
0.6448 (0.6377)
4.66 (5.05)
0.12
4.14
0.05
4.61b
0.13a
4.34a
0:06a
IrMnAl
0.5915 (0.5992)
4.09 (4.0)
0.17
3.70
0.02
IrMnGa
0.5930 (0.6027)
4.08
0.13
3.73
0.01
PdMnTe
0.6336 (0.6271)
4.93 (4.8)
0.27
4.19
0.10
RuMnGa
PtMnGa
0.5695 (0.6150)
0.5986 (0.6150)
3.97
4.04 (3.18)
0.10
0.10
3.62
3.92
0.03
0:07
0.03
3.79
0:10
PtMnSn
0.6204 (0.6263)
3.69 (3.65)
CoTiSb
0.5886 (0.5884)
0.0
CoVSb
0.5821 (0.5801)
1.0 (1.26)
0:20
1.12
0:03
0.97a
0:13a
1.07a
0:02a
0
b
Ref. 30); Ref. 39).
the details in Ref. 30). As in the full-Heusler alloys the
minority occupied states are mainly of Ni ones while the
unoccupied states are of Mn d-states. It is believed that s-p
elements play very important role in the context of existing
energy gap at EF . The replacement of Sb by Al, Ga, Sn or Te
modifying the total magnetic moment in different way but
often the simple rule M = Z-18 is worked.30) The magnetic
moment of PdMnTe is significantly larger than that in
PdMnSb. It is known that Pd is very sensitive to lattice
expansion. As seen from Table 2 the value of spin magnetic
moment is increased significantly in PdMnTe alloys. The
replacement of Sb by Sn in PtMnZ alloys leads to decrease of
magnetic moment from 4.0 down to 3.69 B . The decrease in
magnetic moment is less that in accordance of the rule
mentioned above. In this case the value of local magnetic
moment at the Pt atom changes significantly less than it is at
the Ni30) or Pd atoms. More significant effect is the shift of
the minority bands by 1:23 1019 J (0.47 eV) above the
Fermi level in spite of that the lattice constant is slightly large
in alloy with Sn but the number of valence electrons
decreases by one in this case. So, the substitution of the s-p
element can changes the number of the valence electrons that
along with the distortion of the lattice can lead to a shift of the
Fermi level with respect to the energy gap. The effect of
spin–orbital interaction is more significant in half-Heusler
alloys compared to full-Heusler alloys because it can lead to
splitting of the bands near the Fermi level and shift of
minority states below the energy gap.28) Actually the orbital
moment is small with respect to the spin magnetic moment.
As was shown in in Ref. 38) only in the case of IrMnSb
alloys it reaches to 0.1 B .
In addition we would like also to note that the substantial
change of the magnetic moment can be obtained if Mn is
replaced but V or Ti in the half-Heusler alloys. Our
calculations showed that CoTiSb is a semiconductor in good
agreement with result of Ref. 30) and CoVSb follows SP
curve and has a total magnetic moment of 1.0 B . The
splitting of V spin subbands is substantial less in comparison
with the Mn bands.
In summary, using full-potential augmented-plane-wave
method we performed first-principles calculations of total
energy, spin density of states for several series of Co-, Fe-,
Mn-based Heusler alloys with composition X2 YZ and XYZ.
We found that the equilibrium lattice constants and calculated total and local magnetic moments are in good agreement with available experimental data. It was shown that in
the full-Heusler alloys the magnetic moment at the X site
depends on s-p element at the Z site by changing of lattice
parameter concerned with different size of Z atom. In
considered alloys large total magnetic moment is determined
by element at the Y site but its local moment can be reduced
due to strong hybridization with the transition metal atom at
the X site. Moreover the X–Y interaction is very important for
alloys with sizable magnetic moment at the X site (Co, Ni,
Pd, etc.). In Ni2 MnZ alloys the Ni contribution is determined
the magnetic behavior of alloys upon tetragonal distortion
8
0
-4
-8
12
8
-12
-8 -4
0
Energy, E / eV
RhMnSb
4
EF
4
0
-4
-8
-12
12
8
-12
-8 -4
0
Energy, E / eV
IrMnSb
4
EF
4
0
-4
-8
-12
-12
-8 -4
0
Energy, E / eV
Fig. 5
4
12
8
NiMnSb
605
EF
4
0
-4
-8
-12
N(E) /(states/eV/cell)
N(E) /(states/eV/cell)
EF
4
-12
N(E) /(states/eV/cell)
CoMnSb
N(E) /(states/eV/cell)
12
12
8
-12
-8 -4
0
Energy, E / eV
PdMnSb
4
EF
4
0
-4
-8
-12
N(E) /(states/eV/cell)
N(E) /(states/eV/cell)
The Electronic Structure and Magnetic Properties of Full- and Half-Heusler Alloys
12
8
-12
-8 -4
0
Energy, E / eV
PtMnSb
4
EF
4
0
-4
-8
-12
-12
-8 -4
0
Energy, E / eV
4
Total spin densities of states for two series of XMnSb alloys.
while Co is responsible for this effect in the Fe2 CoGa alloy.
The Co deficiency in the Co2x ZrSn alloy leads to substantial
change in both total and local Co magnetic moment near the
vacancy but this effect is less expressed in the case of
Ni2x MnGa alloy where Mn is responsible for large magnetic
moment. The decrease of the Ni concentration does not lead
to significant change of the magnetic moment at the Mn site
due to its localized character. The calculation of XMnZ halfHeusler alloys exhibits also the dependence of the magnetic
properties on the Z element. Our results confirm that NiMnSb
and PtMnSb have half-metallic behavior. In the semirelativistic approach the highest minority state is located
only 0:06 1019 J (0.04 eV) above the Fermi level. In
general the obtained equilibrium lattice constants and
magnetic moments of half-Heusler alloys agree quite well
with available experimental data.
Acknowledgement
This work was supported by the Russian Foundation for
Basic Research under grant 05-02-16074.
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