ELSEVIER
Physica B 210 (1995) 140-148
Magnetic and half-metallic properties of new Heusler alloys
Ru2MnZ (Z = Si, Ge, Sn and Sb)
S. I s h i d a a'*, S. K a s h i w a g i a, S. F u j i i a, S. A s a n o b
aDepartment of Physics, Faculty of Science, Kagoshima University, Kagoshima, 890 Japan
binstitute of Physics, College of Arts and Sciences, University of Tokyo, Meguro-ku, Tokyo, 153 Japan
Received 19 August 1994
Abstract
Electronic structures of new Heusler alloys Ru2MnZ (Z = Si, Ge, Sn and Sb) were calculated to examine the magnetic
properties. In this paper, it will be shown that an antiferromagnetic state is stable for these alloys, where the Mn magnetic
moments are aligned ferromagnetically in the (1 1 1) plane and antiparallel along the [1 1 1] direction. It will also be
shown that the alloys Ru2MnZ (Z = Si, Sb) can be half-metallic in a ferromagnetic phase.
1. Introduction
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 manganese Heusler
alloys. X2MnZ have been calculated and it was
found that the magnetic moment is mainly carried
by the Mn atoms except for the alloys with X = Co.
Most of X2MnZ are ferromagnets but a few alloys
exhibit antiferromagnetism. Webster and Ramadan
[1] showed that as Sb and Sn substitute in the
alloys Pd2MnSbxlnl_x and Pd2MnSn~Inl_~, the
magnetic order changes an antiferromagnetic type
2 (af2) to a type 3 (af3) and then to a ferromagnetic
(f) order which are shown in Fig. 1. Kiibler et al. [2]
compared the total energies for the f, afl and af2
magnetic phases and showed that the af2 phase is
* Corresponding author.
stable for Pd2Mnln. Fe2MnSi [3] is antiferromagnetic at low temperatures and it was theoretically
shown by Fujii et al. [4] that the stable state is the
af2 state.
Recently, new Heusler alloys Ru2MnZ (Z = Si,
Ge, Sn and Sb) were prepared by Kamomata et al.
I-5] and they showed from the temperature dependence of magnetization that these alloys are antiferromagnetic. Neutron diffraction studies are in progress by Gotoh et al. [6] and the af2 state was
observed for RuEMnGe, Ru2MnSn and Ru2MnSb.
In this situation, we calculated the electronic
structure of Ru2MnZ (Z = Si, Ge, Sn and Sb) and
will shows the magnetic properties obtained theoretically.
2. Crystal structure and method of calculation
The crystal structure of Heusler alloys X2MnZ
is of the L21 type, where Mn and Z atoms are
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 9 2 0 - 1
S. lshida et al. /Physica B 210 (1995) 140-148
ferro
dl
w
d2
d3
Fig. 1. The arrangement of the Mn magnetic moment in
Ru2MnZ. The ferromagnetic arrangement is indicated by f and
the three types of the antiferromagnetic by afl, af2 and af3.
surrounded by eight X atoms. We considered four
magnetic states, i.e. a ferromagnetic (f) and three
antiferromagnetic states (afl, af2 and af3). The arrangements of Mn magnetic moments in these
magnetic states are shown in Fig. 1. The afl and af2
states are characterized by alternating ferromagnetic planes and up- and down-moments perpendicular to the [1 00] and [1 1 1] directions, respectively. In the af3 state, adjacent moments along two
of the cube edges are aligned parallel and along the
other are aligned antiparallel. The number of space
groups for the f, afl, af2 and af3 states are assumed
to be 225, 129, 166 and 141 in the international
table [7].
The energy eigenvalues were determined by the
LMTO-ASA method in the non-relativistic approximation [8] and the exchange-correlation potential is within the framework of the LSD approximation [9]. We prepared several sets of the radii of
the atomic spheres in order to find reasonable equi-
141
librium lattice parameters and the following was
finally used: the ratio of the radii rRu :r~n :rz is
1.15 : 1.0 : 1.05 for Z = Si, 1.15 : 1.0: 1.0 for Z = Ge,
and 1.209: 1.0:1.1 for Z = Sn and Sb. The radii
used here produce reasonable results about the
lattice constant as seen later. The maximum angular momenta to describe the valence states were
chosen to be lmax = 2 for all of atoms. The density of
states was determined by the tetrahedral integration method [10].
3. Total energy and ground magnetic phase
The total energies for Ru2MnZ (Z = Si, Ge, Sn
and Sb) were calculated as functions of the lattice
constant a which were shown in the lower parts of
Fig. 2(a)-(d). The full, dashed, dotted and dasheddotted-dashed curves are for the f, afl, af2 and af3
state, respectively. The difference of the minimum
energy between the f and the other states are listed
in Table 1. It is found from the figures and the
energy differences that the total energy is lowest for
the af2 state and becomes large in the order of af2,
f, afl and af3 for all of the alloys. Really, the af2
states were observed for Z = Ge, Sn and Sb [6].
The theoretical values of the lattice constant were
determined so as to minimize the total energies.
The values are listed in Table 1 and are compared
with the experimental values. The agreement between the theoretical and observed values is reasonable. In the next section, we will see the densityof-state at the theoretical lattice parameter.
We discuss the Heisenberg exchange constants as
Kiibler et al. [2] did. Using the Heisenberg Hamiltonian
I4 = - E J , ~ s , . s j ,
i,j
the difference of the exchange energy is described as
follows:
E = E(ferro) - E(antiferro)
--
4S2EziJi,
i
where Ji is the ith exchange constants and z~ is the
ith coordination number for oppositely ordered
moments. In these calculations, we neglect the
S. lshida et al. / Physica B 210 (1995) 140-148
142
Table 1
The lattice parameter a, the magnetic moment and the energy difference (in Ryd) for Ru2MnZ (Z = Si, Ge, Sn and Sb). The experimental
(EX) values are also listed. The moments were determined by neutron diffraction experiments (ND) and the Curie-Weiss law (CW)
Z
Magnetic phase
a(A)
Moment (/IB)
Mn
Si
f
afl
af2
af3
5.95
5.95
5.96
5.95
3.11
3.16
3.20
3.16
EX
5.887
f
afl
af2
ag3
6.05
6.05
6.06
6.05
EX
5.985
f
afl
af2
af3
6.30
6.30
6.30
6.30
EX
6.217
f
afl
af2
af3
6.30
6.30
6.30
6.30
EX
6.200
Ge
Sn
Sb
Ear - Ef
- 0.06
0.0047
- 0.0020
0.0060
0.00
2.83(CW)
3.15
3.20
3.23
3.20
- 0.07
0.0042
- 0.0025
0.0055
- 0.05
3.8(ND)3.17(CW)
3.43
3.52
3.56
3.53
- 0.20
0.0015
- 0.0061
0.0021
- 0.05
4.2(ND) 2.81(CW)
3.72
3.80
3.81
3.79
0.14
0.0044
-- 0.0048
0.0050
0.10
4.4(ND) 3.87(CW)
moments on the Ru and Z atoms, and the exchange
constants Ji between Mn moments were determined up to i = 3. The estimated values are listed in
Table 2. The first neighbor interaction (J1 > 0) is
ferromagnetic but the second and third interactions
(J2 < 0, J3 "(0) are
Moment (#B)
Ru
antiferromagnetic. The ground
phase diagram obtained by Moran-Lopez et al.
[11] are shown in Fig. 3 for two cases of J1 > 0 and
J3 < 0. The
coordinates corresponding to
Ru2MnZ (Z = Si, Ge, Sn and Sb) are also shown by
the symbol of cross, circle, square and triangle,
respectively, which are in the region of the af2
phase. The symbols for Z = Si and Ge are located
near the boundary between the f and the af2 phase
and there is a tendency that the af2 phase becomes
stable with increasing the atomic number of the
Z atom. Thus, we can predict from the total energy
and the ground phase diagram of exchange constants that the stable magnetic phase is the af2 state
for all of our interested alloys.
We also estimated Neel temperatures TN which
are described by a formula
kBTN =
-
4S(S +
1)J2,
where kB is Boltzman's constant. We assumed for
S the value of the magnetic moment on the Mn
143
S. lshida et al./Physica B 210 (1995) 140 148
#
I
I
I
I
I
I
I
I
I
I
af2
afl
af3
--='3.3
I
I
I
I
I
I
El
I
I
~3.3
ferro
h,
°
I
af2
af3
all
...~.~j~
•*'~'~
ferro
=IE
I
~3.1
o=2.9
~2.9
I
I
I
\
I
I
I
I
I
I
I
Ru2HnSi
I
I
1•\
10mR%d
I
I
I
I
I
I
I
I
\
•.
10mRyd I
\.
%.?
>-
IX=
I.i
Z
I.s-I
I
Ru2HnG~
Z
1.1..I
..-I
~
I
(a)
I
I
I
ferro
af2
"
I
I
I
I
I
I
I
I
I
I
I
.
af 1
ferro
af2
I
I
5.80
5.90
6.00 6.10
LATTICE CONSTANT (A)
5.70
' " - ~lt>-_.--.~W
......
I-.--
I
I
5.80
(b)
I
I
I
I
I
I
I
5.g0
6.00
6.10
6.20
LATTICE CONSTANT (AI
I
af2
I
I
I
I
I
#
I
I
I
I
I
af3
--="3.6
-=-3.9
all
~: 3.4-
~3.7
N3.2
af2
af I
.~.~i
ferro
. ~_..>~>
fcrro
Lid
z3.5
3=
IX2
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
Ru2MnSn
10mRyd
¢...~\
/~
IOmFlyd
¢..0
ta.J
2Z
t----
I
(c)
6.10
I
t
I
t
I
taJ
af3
"""!-.
""~ ~
----~'~"
....... ~ ......
• .
afl
.
""
ferro
af2
I
I
6.20
6.30
6.¢0
6.50
LATTICE CONSTANT IAI
.
0
.
.
~
•
.
.
0'"
"''"''0'''"'"
af3
afl
ferro
af2
t
I
(d)
6.10
I
I
I
I
I
I
I
I
6.20
6.30
6.¢0
6.50
LATTICE CONSTANT IAI
Fig. 2. The lattice constant dependence of the total energy per formula unit and the magnetic moment for (a) Ru2MnSi, (b) Ru2MnGe,
(c) Ru2MnSn and (d) Ru2MnSb. The full, dashed, dotted and dashed-dotted-dashed curves distinguish the ferro, afl, af2 and af3,
respectively.
144
S. lshida et al./Physica B 210 (1995) 140-148
Table 2
The exchangeconstants(in meV)and the Neel temperature(TN)
for Ru2MnZ(Z = Si, Ge, Sn, Sb).The experimentalvalues were
determined by neutron diffractionexperiment (ND) and the
Curie-Weiss law (CW)
Z
Si
Ge
Sn
Sb
Jl
J2
0.339
0.323
0.186
0.239
J3
- 0.229
-0.226
-0.238
-0.233
U 31
0
i
TN
- 0.096
-0.098
-0.074
-0.071
i
cal
ND
CW
194
195
244
270
320
315
200
313
316
296
195
i
t
i
2.0
dependence of the susceptibility by Gotoh et al. [6]
and Kanomata et al. [12]. The theoretical value
is smaller than the experimental value except for
Ru2MnSb. One of the reasons of the discrepancy
is attributed to the discrepancy of the magnetic
moments between the theoretical and the experimental value described later. Gotoh et al. I-6] observed the following results about the direction of
the Mn moments. The Mn moments are along the
[1 1 1] direction in Ru2MnGe and Ru2MnSn.
However, in Ru2MnSb the Mn moments make an
angle of 35 ° from the 1-1 1 l] direction at low temperatures and the Mn moments rotate to the 1-1 1 1]
direction with increasing temperatures. Thus the
behavior of the Mn moments in Ru2MnSb is peculiar compared with the other alloys. This feature
may be related with the Neel temperature and
further investigation is needed to obtain a conclusive results.
0
..-)
-b
-2.0
4. Density of state
af2
-4-.0
'
-,i0
'
,,~
JI /IJ3 I
i
'
...-'o
i
'
i
J;>O
0.5
f
o
-) -0.5
-1.0
a{2
af I
i
-1.0
0
J2/J;
1.0
Fig. 3. The ground phase diagram of the exchange constants for
the cases of Jr > 0 and J3 < 0. The coordinates indicated by the
symbols of cross, circle, square and triangle correspond to
Ru2MnSi, Ru2MnGe, Ru2MnSn and Ru2MnSb.
atom in the af2 phase. They are compared in Table
2 with the experimental values determined by neutron diffraction experiments and the temperature
We first pay our attention to the density-ofstate (DOS) of a ferromagnetic state of Ru2MnSi.
The local DOSs are shown for the d-states of the
Ru and Mn atoms and the p-states of Si atoms in
Fig. 4. The DOS curves of up (majority)-spin and
down (minority)-spin are drawn by full and dotted
lines, respectively, and the Fermi level by a
vertical line. The valence states of Ru, Mn and Si
atoms well hybridize each other in wide energy
range. The local DOS of Mn d-states shows that
the d-bands of both spin states are widely separated
and this separation brings a large magnetic moment at Mn sites. This separation of Mn d-bands
affects the other valence states. That is, the
hybridization between the Mn d-states and the
other valence states produces differences in local
DOSs of both spin states. Especially, the difference
between up- and down-spin states is remarkable in
the DOSs of Ru d-states. For example, the DOS of
the up-spin state has peaks near the Fermi level
while there is a gap near the Fermi level in the DOS
of the down-spin state. However, the spin polarization of Ru atom is negligible small as shown in
Table 1.
145
S. Ishida et al./Physica B 210 (1995) 140-148
p
60
i
i
i
i
Mn (d)
40
i
ij
20
_
30
i"~ (d)
2oi
40
-&
af3
Ru (d)
40
af2
~ 2oi
--~ 2O
r"r"
cr
) lo
)40
-
-
afl
i:
O"3
15
Si
(p)
lO
-0.3
(11
0.3
-0.1
Energy (Ryd)
there is not an energy gap of the d o w n - s p i n b a n d
w h i c h is seen for the f phase. As seen above, the
difference for four m a g n e t i c phase appears clearly
in the local D O S o f the R u d-states.
]
20
J!.,.....
-0.5
(a)
Fig. 4. The local DOS curves of the d states of Ru and Mn
atoms and the p-states of Si atom in Ru2MnSi. The full and
dotted curves show the DOS of up-spin and down-spin states,
respectively. The Fermi level is shown by a vertical line.
We next see the DOSs of three antiferromagnetic
states (afl, af2 and af3) of Ru2MnSi. The local
DOSs of the Mn d-states are shown in Fig. 5(a) and
those of the Ru in Fig. 5(b). Comparing the DOS
shapes of four magnetic phase, we find that they are
similar each other for the Mn atoms but different
for the Ru atoms. The Ru atom in the afl phase and
the Z atom in the af2 phase are nonmagnetic due to
the symmetry, i.e. their DOS of the up- and downspin states are equivalent. In Fig. 5(b), the local
DOS of one of two kinds of the Ru atoms is shown
for the af2 phase. As described above, the DOSs of
both spin states are different for the f phase and the
values of DOS for the afl and af3 phases are nearly
equal to the average values of those for the up- and
down-spin states in the f phase. In the af2 phase,
40 ferro Iv~q(d)
-0.3 -0.1
Energy (Rydl
0.1
0.3
-0.3 -0.1
0.1
Energy CRyd)
0.3
20
af3
Ru(d)
20
af2
P,u(d)
20t all
Ru(d)
cr
20t ferro F:{a[d)
-0.5
(b)
g~i
Fig. 5. The local DOS curves of the f, afl, af2 and af3 states for
(a) the Mn d-states and (b) the Ru d-states in Ru2MnSi.
146
S. lshida et al. / Physica B 210 (1995) 140-148
The DOSs of the other alloys with Z = Ge, Sn
and Sb are similar to those of Ru2MnSi. The DOSs
shown in Figs 6(a) and (b) are of Ru2MnSb where
the number of the sp electrons of the Sb atom is one
larger than that of Si atom. The detail structures
are different from those of Ru2MnSi but the general
features are similar. From the position of the Fermi
level, we notice that the up-spin states are more
occupied in Ru2MnSb than in Ru2MnSi for all of
the magnetic phases. Therefore, it is expected that
the magnetic moment on the Mn atoms is larger in
Ru2MnSb than in Ru2MnSi.
':: i
40
af3
Mn(d}
i:
20
4-0
"!
Mn{dl~~
a{2
2~
20~
rr-
)40
all
~ 2O
5. Magnetic moment
4-0
f e r r u Pin[d)
20
-0.5
-0.3
-0.1
Energy' (Ryd)
(a)
20
a~t:3
Ru Id)
af2
Ru (d}
0.1
0.3
1r
10
20
~
~-I0
.
-=
,
.
:
~
20
i
20 ¸
.
.
.
.
f e r r o Ru(d)
~.
i
i
i
,::
10.
-0.5
(b)
-0.3
-0.1
Energy (Ryd)
0.1
0.3
Fig. 6. The local D O S curves of the f, afl, af2 and af3 states for
(a) the Mn d-states and (b) the Ru d-states in Ru2MnSb.
The magnetic moments on Mn sites are plotted
in the upper part of Figs. 2(a)-(d) as functions of the
lattice constant a. The magnetic moment increases
monotonically with the lattice expansion for all of
the magnetic phases. The full, dashed, dotted and
dashed-dotted-dashed curves distinguish the magnetic phases as for total energy. The moment is
smallest for the f phase, largest for the af2 phase
and those for the afl and af3 phase are nearly equal.
The local moments at the lattice constant which
produces the minimum energy are summarized
in Table 1. The experimental values determined
by neutron diffraction experiments and the CurieWeiss law [6, 12] are also shown in the same table.
As expected from the similarity of the Mn DOSs, the
Mn local moments are nearly equal for the four
magnetic phases. The experimental and theoretical
values of the moment become larger with increasing
the atomic number of the Z atom (Si, Ge, Sn and
Sb). The Ru local moments are small and are antiparallel to the Mn moment except for the case of
Z = Sb. The theoretical values are smaller compared
with the experimental values as other results for
X2MnZ (X = Cu, Ni and Pd; Z = A1, Sn and Sb)
[2, 13]. The reason of the difference is not clear. One
of the reasons may be attributed to the fact that the
Ru sites of 30% in Ru2MnSn used for experiments
are occupied by Sn atoms and the Ru sites of 7% in
Ru2MnSb by Sb atoms. The refinement of the samples and a good approximation for the calculation of
the crystal potential are needed to discuss quantitatively the magnetic moment and the Neel temperature.
S. lshida et al. / Physica B 210 (1995) 140-148
6. Half metalic alloys
Ru 2 ["In Sb
As seen in the DOS of a ferromagnetic Ru2MnSi,
the Fermi level is positioned in an energy gap of the
down-spin band but at a DOS peak of the up-spin
band. Therefore, it is expected that Ru2MnSi is half
metallic in the f magnetic phase• That is, this alloy
shows normal metallic behavior for the up-spin
while at the same time being an insulator or
semiconductor for the down-spin.
To clear it, we show the energy dispersion curves
(E(k) curve) near the Fermi level• Fig. 7 is for
Ru2MnSi, where the upper is for the minority
(down)-spin and the lower for the majority (up)spin. The Fermi level is shown by a horizontal
dotted-line. The energy curves cross the Fermi level
in the up-spin state but the Fermi level is positioned
just above the top of the valence band in the downspin state• The E(k) curves of the down-spin state
are shown in Fig. 8 for other alloys with Z = Ge, Sn
and Sb. The Fermi level is just below the top of the
valence band for Z = Ge and Sn while above for
Z = Sb. Thus, our results suggest that among the
alloys Ru2MnZ (Z = Si, Ge, Sn and Sb), the alloys
with Z = Si and Sb can be half-metallic in the f magnetic phase. Fe2MnSi, C02MnZ (Z = Si, Ge) can
MINORITY
147
>132
Z-.1
LI
Ru 2 Nn Sn
t.~z- - 1
Ru z Nn Ge
F
A
XZW
Q
L
A
Z
K
X
Fig. 8. The energy dispersion curves of the down-spin state for
Ru2MnGe, Ru2MnSn and Ru2MnSb.
also be half-metalic. Their results will be reported
elsewhere•
SPIN
7. Summary
>.C9
O=
LLJ-Z
"
LU
MAJORITY
SPIN
E
>.(..9
r'r"
1,4JZ
"
2
A
XZW
Q
L
A
F
}:
K
Fig. 7. The energy dispersion curves near the Fermi level for
Ru2MnSi. The Fermi level is shown by a dotted line.
The electronic structures of new Heusler alloys
Ru2MnZ were calculated to examine the magnetic
properties for a ferromagnetic and three antiferromagnetic phases. It was found that the most
stable phase for all of the alloys is the af2 phase
where the Mn magnetic moments are aligned ferromagnetically in the (1 1 1) plane and are antiparallel in the [1 1 1] directions. This result is consistent with the ground phase diagram of exchange
constants obtained by Moran-Lopez et al. [11].
The af2 phase was really observed in Ru2MnGe,
Ru2MnSn and Ru2MnSb. The characteristics of
the electronic structures for the four magnetic
phases appear in the local DOS of the Ru atom.
The theoretical values are in good agreement with
the experimental values for the lattice parameters
148
S. lshida et al. / Physica B 210 (1995) 140-148
but the difference between the theoretical and the
experimental value is not negligible for the magnetic moments and the Neel temperatures. Further
investigations must be done theoretically and experimentally to discuss quantitatively.
It was also found that Ru2MnZ (Z = Si, Sb) can
be half-metallic in the ferromagnetic phase.
Acknowledgements
We would like to thank Mr. Tsuyoshi Takiguchi
for help of parts of the calculations. This work was
partially supported by a Grant-in-Aid for Scientific
Research from the Ministry of Education, Science
and Culture of Japan.
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