Materials Transactions, Vol. 57, No. 3 (2016) pp. 312 to 315
© 2016 The Japan Institute of Metals and Materials
Atomic Arrangement and Magnetic Order in Mn2RuZ (Z = Sn, Si)+1
Koki Shimosakaida+2 and Shinpei Fujii
Graduate School of Science and Engineering, Kagoshima University, Kagoshima 890-0065, Japan
Recently, new compounds Mn2RuSn and Mn2RuSi were synthesized and their crystal structures were studied by X-ray diffraction
measurements. This measurement indicates that they have a Heusler-like cubic structure, but the details have been unclear so far. To clarify their
atomic arrangement and magnetic order, we carried out first-principles total-energy calculations for several different atomic arrangements,
together with ferrimagnetic and ferromagnetic ordering. The comparison among total energies indicates that the most stable structure is a
ferrimagnetic Hg2CuTi one in both Mn2RuSn and Mn2RuSi. We also found that the compound Mn2RuSi with the Hg2CuTi structure could be a
half-metallic ferrimagnet. [doi:10.2320/matertrans.M2015358]
(Received September 25, 2015; Accepted December 25, 2015; Published February 15, 2016)
Keywords: Mn2RnZ, Hg2CuTi, first-principles calculation, ferrimagnetic, half-metallic
1.
Recently many compounds with a type of Mn2YZ are
synthesized though Mn2VAl is the only Heusler compound
30 year ago.1) For example, there are Mn2NiGa,2,3) Mn2CoZ
(Z = Al, Ga, In, Si, Ge, Sn, Sb),4) and Mn2RuZ (Z = Si, Sn).
However, they do not always have a crystal structure of L21
(Fm3m)
(see Fig. 1). Liu et al.2) synthesized Mn2NiGa and
showed that the compound undergoes a structural phase
transformation from a cubic austenite phase to a tetragonal
martensite phase with decreasing temperatures. From X-ray
diffraction measurements, they also reported that the high
temperature parent phase is not a L21-type structure but a
Hg2CuTi-type one (F 43m).
However, they did not consider
the possibility of a L21B-type structure (see Fig. 1). Recently,
Brown et al. reported that Mn2NiGa has a L21B-type
structure at the high temperature parent phase from neutron
powder diffraction measurements. Concerning Mn2CoZ
(Z = Al, Ga, In, Ge, Sn, Sb), it was reported that they have
not a L21B-type structure but a Hg2CuTi-type one from X-ray
diffraction measurements.4) Furthermore, it is predicted that
the compounds Mn2CoZ (Z = Al, Si, Ge, Sn, Sb) are halfmetallic ferrimagnets from first-principles calculations.4)
Endo et al.5) showed that Mn2RuZ (Z = Sn, Si) crystallizes in a Heusler-like cubic structure (L21B- or XA-type
structure). The lattice constants of Z = Sn and Si at room
temperature are estimated to be 6.2195 ¡ and 5.8260 ¡,
respectively. However, there is no investigation from firstprinciples calculations to determine atomic arrangement in
Mn2RuZ (Z = Sn, Si). In this paper, we report the results of
first-principles total-energy calculations to clarify the structural and magnetic properties of Mn2RuZ (Z = Sn, Si).
2.
Approach
We consider four structural types as shown in Table 1,
which are adopted to reveal the atomic arrangement at the
high temperature parent phase of Mn2NiGa.3) Here, L21A and
XA are well known as L21 and Hg2CuTi, respectively and
+1
(a)
Introduction
This Paper was Originally Published in Japanese in J. Japan Inst. Met.
Mater. 79 (2015) 210­214.
+2
Graduate Student, Kagoshima University. Corresponding author, E-mail:
[email protected]
(b)
Z
Y
Z
Mn
(c)
Mn
1/2 Y,1/2 Mn
(d)
Z
Mn
Y
Z
2/3 Mn,1/3Y
Fig. 1 Several crystal structure of Mn2YZ: (a) L21 (L21A), (b) L21B,
(c) XA, (d) DO3.
Table 1 Different atomic arrangements in Mn2YZ. For example, the word
“(1/2Y, 1/2Mn)” on the site C of L21B means that the Y and Mn atoms
have an equal occupation probability of the site C.
structure
A
(0, 0, 0)
B
ð1=2; 1=2; 1=2Þ
C
ð1=4; 1=4; 1=4Þ
D
ð3=4; 3=4; 3=4Þ
L21A
Z
Y
Mn
Mn
L21B
Z
Mn
(1/2Y, 1/2Mn)
(1/2Y, 1/2Mn)
XA
DO3
Z
Z
Mn
Y
Mn
(2/3Mn, 1/3Y) (2/3Mn, 1/3Y) (2/3Mn, 1/3Y)
each of them is an ordered structure. On the other hand, L21B
and DO3 are structures with an atomic disorder. The
expression as “(2/3Mn, 1/3Y)” for B, C, and D sites in
DO3 in Table 1, for example, means that the Mn and Y atoms
randomly occupy each site with the ratio of 2 to 1.
Concerning magnetic ordering, we consider two types as
ferromagnetic, and ferrimagnetic states. For each structural
type and magnetic ordering, we estimate its total energy and
determine the most stable state comparing their total energies.
First-principles total-energy calculations are performed
using the full-potential linearized augmented plane wave
(FLAPW) method (WIEN2k package).6) The generalized
gradient approximation of Perdew et al.7) is used for the
exchange-correlation potential. The plane wave cutoff is
RKmax = 7.0, Where R is the smallest atomic sphere radius
and Kmax is the magnitude of the largest K vector. The
Atomic Arrangement and Magnetic Order in Mn2RuZ (Z = Sn, Si)
Fig.2.1
(a)
Table 2 Lattice constant amin (¡) and magnetic moment (®B) of constituent
atoms and cell (formula unit) for Mn2RuSi.
(b)
Energy per formula unit, E/eV
WIEN2k
L21A
amin
0.2
5.8
6.0
6.2
5.6
5.8
L21A
XA
6.0
Lattice constant, a/Å
(a)
Energy per formula unit, E/eV
0.2
L21A
XA
6.2
6.4
6.6
6.0
6.2
6.4
6.6
Lattice constant, a/Å
Fig. 2 Energy vs. lattice constant for Mn2RuSi(2.1), Mn2RuSn(2.2)
calculated by (a) WIEN2k and (b) KKR. The symbols
and
mean
the results of L21A- and XA-type crystal structures, respectively.
following atomic sphere radii are used: 2.24 a.u. (2.40 a.u.) for
Mn and Ru, and 2.11 a.u. (2.25 a.u.) for Si (Sn) in Mn2RuSi
(Mn2RuSn). We also use the Korringa-Kohn-Rostoker (KKR)
method (Akai KKR package)8,9) because this package
provides the coherent potential approximation (CPA)10)
program to deal with an atomic disorder (KKR-CPA).
For the initial value for a lattice constant, we used the
experimental values, a = 5.826 ¡ (6.2195 ¡) of Mn2RuSi
(Mn2RuSn).5) We calculated total energies against five or six
lattice constants and fitted an energy-vs-volume function using
the Murnaghan equation of state.7) Then, we estimated the value
of the lattice constant amin (or the volume Vmin), which gives a
minimum of the function. The ground-state properties (the
magnetic moments, density of states and so on) are re-estimated
at the value amin for each structural type and magnetic ordering.
Results and Discussion
We show an energy-vs-volume curve for Mn2RuZ (Z = Si
and Sn) with a L21A- or XA-type structure in Fig. 2. The
obtained data (amin and magnetic moments of the constituent
atoms) are listed in Tables 2 and 3. Comparing between two
results of WlEN2k and KKR, we notice the following
aspects. The XA-type structure is more stable than the L21A
in both of WIEN2 and KKR; The value of amin of the XAtype structure is smaller than that of the L21A in both; The
value of the magnetic moment per cell is almost same in both
except for XA of Mn2RuSn, where 1.9 ®B (WIEN2k) and
0.9 ®B (KKR), and so the difference is not small. From these
facts, we confirm that the present KKR package gives similar
results to the WIEN2k package in basic physical properties,
where the latter is more accurate than the former because the
latter makes no shape approximation to the potential or the
electron density but the former does not. Hereafter, we show
the results of KKR-CPA and discuss them.
5.801
2.834
Ru
0.068
ð3=4; 3=4; 3=4Þ
Mn
2.834
Mn
¹0.809
ð1=2; 1=2; 1=2Þ
Ru
0.827
Mn
2.658
(0, 0, 0)
Si
¹0.071
Si
0.027
cell
6.585
cell
2.002
L21A
amin
(b)
6.0
Mn
KKR
0.2
5.8
ð1=4; 1=4; 1=4Þ
6.2
Fig.2.2
XA
5.937
XA
6.010
ð1=4; 1=4; 1=4Þ
Mn
ð3=4; 3=4; 3=4Þ
Mn
ð1=2; 1=2; 1=2Þ
(0, 0, 0)
Ru
Si
cell
7.021
5.870
3.115
Ru
0.095
3.115
Mn
¹1.248
0.796
¹0.075
Mn
Si
3.062
0.045
cell
2.001
Table 3 Lattice constant amin (¡) and magnetic moment (®B) of constituent
atoms and cell (formula unit) for Mn2RuSn.
WIEN2k
L21A
amin
XA
6.292
6.181
ð1=4; 1=4; 1=4Þ
Mn
3.407
Ru
0.271
ð3=4; 3=4; 3=4Þ
Mn
3.407
Mn
¹1.913
ð1=2; 1=2; 1=2Þ
Ru
0.964
Mn
3.398
(0, 0, 0)
Sn
¹0.050
Sn
0.031
cell
7.841
cell
1.889
L21A
KKR
amin
XA
6.357
6.334
ð1=4; 1=4; 1=4Þ
Mn
3.522
Ru
0.133
ð3=4; 3=4; 3=4Þ
Mn
3.522
Mn
¹3.151
ð1=2; 1=2; 1=2Þ
(0, 0, 0)
Ru
Sn
0.859
¹0.067
Mn
Sn
3.785
0.048
cell
7.807
cell
0.879
(a)
(b)
0.1
Energy per formula unit, E/eV
0.4
5.6
3.
313
5.6
L21A
L21B
XA
DO3
0.2
5.8
6.0
6.2 6.0
6.2
6.4
6.6
Lattice constant, a/Å
Fig. 3 Energy vs. lattice constant for Mn2RuSi(a), Mn2RuSn(b) calculated
by KKR-CPA. The symbols , , , and mean the results of L21A-,
L21B-, XA-, and DO3-type crystal structures, respectively.
Figure 3 shows an energy-vs-volume curve of Mn2RuZ
(Z = Si and Sn) for the four structures such as L21A, XA,
L21B, and DO3. The obtained data (amin and magnetic
moments of the constituent atoms or “pseudo atoms”) are
listed in Tables 4 and 5, where “pseudo atom”, for example,
means “(2/3Mn, 1/3Y)” for B, C, and D sites of DO3 in
Table 1. In addition to Mn2RuZ (Z = Si and Sn), we have
performed the same calculation for the high temperature
314
K. Shimosakaida and S. Fujii
Table 4 Lattice constant amin (¡) and magnetic moment (®B) of constituent
atoms and cell (formula unit) for Mn2RuSi.
Table 6 Lattice constant amin (¡) and magnetic moment (®B) of constituent
atoms and cell (formula unit) for Mn2NiGa.
KKR
L21A
XA
L21B
DO3
WIEK2k
amin
6.010
5.870
5.942
5.998
amin
L21A
ð1=4; 1=4; 1=4Þ
3.115
0.095
¹0.701
1.272
ð1=4; 1=4; 1=4Þ
Mn
3.248
Ni
0.336
ð3=4; 3=4; 3=4Þ
3.115
¹1.248
¹0.701
1.272
ð3=4; 3=4; 3=4Þ
Mn
3.248
Mn
¹2.337
ð1=2; 1=2; 1=2Þ
0.796
3.062
3.26
¹1.417
ð1=2; 1=2; 1=2Þ
Ni
0.678
Mn
3.128
(0, 0, 0)
¹0.075
0.045
0.06
¹0.077
(0, 0, 0)
Ga
¹0.068
Ga
0.012
cell
7.021
2.001
1.971
1.047
cell
7.192
cell
1.186
5.841
L21A
KKR
Table 5 Lattice constant amin (¡) and magnetic moment (®B) of constituent
atoms and cell (formula unit) for Mn2RuSn.
XA
5.943
amin
XA
5.990
5.940
ð1=4; 1=4; 1=4Þ
Mn
3.354
Ni
0.211
KKR
L21A
XA
L21B
DO3
ð3=4; 3=4; 3=4Þ
Mn
3.354
Mn
¹2.879
amin
6.357
6.334
6.368
6.387
ð1=4; 1=4; 1=4Þ
3.522
0.133
¹1.628
2.259
ð1=2; 1=2; 1=2Þ
(0, 0, 0)
Ni
Ga
0.650
¹0.088
Mn
Ga
3.452
0.010
ð3=4; 3=4; 3=4Þ
3.522
¹3.151
¹1.628
2.259
cell
7.272
cell
0.792
ð1=2; 1=2; 1=2Þ
(0, 0, 0)
0.859
¹0.067
3.785
0.048
3.72
0.037
¹2.301
¹0.047
cell
7.807
0.879
0.518
2.153
(b)
Energy per formula unit, E/eV
(a)
5.6
0.2
L21A
L21B
XA
0.2
DO3
Table 7 Lattice constant amin (¡) and magnetic moment (®B) of constituent
atoms and cell (formula unit) for Mn2NiGa.
KKR
L21A
XA
L21B
DO3
amin
5.990
5.940
5.879
5.894
ð1=4; 1=4; 1=4Þ
3.354
0.211
¹0.856
1.175
ð3=4; 3=4; 3=4Þ
3.354
¹2.879
¹0.856
1.175
ð1=2; 1=2; 1=2Þ
0.65
3.452
3.322
¹1.317
(0, 0, 0)
cell
¹0.088
7.272
0.01
0.792
0.015
1.644
¹0.044
0.965
Table 8 Lattice constant amin (¡) and magnetic moment (®B) of constituent
atoms and cell (formula unit) for Mn2VAl.
5.8
6.0
6.2 5.6
5.8
6.0
6.2
Lattice constant, a/Å
Fig. 4 Energy vs. lattice constant for Mn2NiGa(a), Mn2VAl(b) calculated
by KKR-CPA. The symbols , , , and mean the results of L21A-,
L21B-, XA-, and DO3-type crystal structures, respectively.
parent phase of Mn2NiGa and shows the results in Fig. 4(a),
Tables 6 and 7. In all compounds, Mn2RuSi, Mn2RuSn, and
Mn2NiGa, a ferromagnetic state is stabilized for L21A and a
ferrimagnetic state is stabilized for the others.
Here, we refer to Mn2VAl because the compound has a
L21A-type structure as a ground-state structure. As shown in
Fig. 4(b), our result is consistent with the experimental one.
The obtained data (amin and magnetic moments of the
constituent atoms or “pseudo atoms”) are listed in Tables 8
and 9.
Our results for Mn2RuSi, Mn2RuSn, and Mn2NiGa (parent
phase) from KKR-CPA show that the XA-type structure has
the lowest energy among L21A, XA, L21B, DO3. Thus, we
show the results of WIEN2k for XA, together experimental
data in Table 10. The table shows that the theoretical value of
a lattice constant is in good agreement with the experimental
one because the difference between them is less than 2%. By
the way, the experimental values listed in Table 10 are
estimated at room temperature for Mn2RuZ (Z = Si and Sn),
500 K for Mn2NiGa, and 4.2 K for Mn2VAl. However, we did
not extrapolate those values to them at T = 0 K to compare the
theoretical values at T = 0 K because the thermal expansion
coefficient of Heusler compounds is about 10¹5 K¹1.11)
WIEK2k
L21A
amin
XA
5.818
ð1=4; 1=4; 1=4Þ
Mn
ð3=4; 3=4; 3=4Þ
ð1=2; 1=2; 1=2Þ
(0, 0, 0)
5.812
1.424
V
¹0.208
Mn
1.424
Mn
0.624
V
¹0.790
Mn
0.603
Al
¹0.017
Al
¹0.017
2.003
cell
cell
L21A
KKR
amin
0.970
XA
5.832
5.901
ð1=4; 1=4; 1=4Þ
Mn
1.457
V
1.315
ð3=4; 3=4; 3=4Þ
ð1=2; 1=2; 1=2Þ
Mn
V
1.457
¹0.820
Mn
Mn
2.114
¹1.638
Al
¹0.029
Al
0.005
cell
1.971
cell
1.968
(0, 0, 0)
Concerning magnetic moments, the error between the
values of WIEN2k and the experimental ones is only 3% for
Mn2NiGa but 10% for Mn2RuSn and Mn2VAl. In the case of
KKR, the error is 47% for Mn2RuSn and 31% for Mn2NiGa.
Thus, we should avoid comparing the experimental data with
the results of KKR.
We make some comments on the electronic structure of
Mn2RuSi. Though it is reported that the compound exhibits
spin-glass-like behavior at low temperature,5) if the transition
can be suppressed and can hold a ferrimagnetic XA-type
structure, then the compound may have a half-metallic
characteristic because the compound has a magnetization of
2.0 ®B/f.u. In fact, the calculated density of states (DOS)
Atomic Arrangement and Magnetic Order in Mn2RuZ (Z = Sn, Si)
Table 9 Lattice constant amin (¡) and magnetic moment (®B) of constituent
atoms and cell (formula unit) for Mn2VAl.
KKR
L21A
XA
L21B
DO3
amin
5.832
5.901
5.874
5.900
ð1=4; 1=4; 1=4Þ
1.457
1.315
1.330
1.576
ð3=4; 3=4; 3=4Þ
1.457
2.114
1.330
1.576
ð1=2; 1=2; 1=2Þ
¹0.820
¹1.638
¹0.243
¹0.010
(0, 0, 0)
¹0.029
0.005
¹0.024
¹1.777
cell
1.971
1.968
1.644
1.417
Table 10 Comparison between calculated and experimental values for
lattice constants and magnetic moments. In the parentheses, the relative
errors (%) are shown.
a(¡)
DOS [States/eV/atom/spin]
DOS [States/eV/f.u./spin]
M(®B)
6
4
2
0
2
4
6
6
4
2
0
2
4
6
6
4
2
0
2
4
6
6
4
2
0
2
4
6
1
exp
WIEN2k
KKR
5)
Mn2RuSi
5.8260
5.801 (0.47%)
5.870 (0.76%)
Mn2RuSn5)
6.2195
6.181 (0.62%)
6.334 (1.8%)
Mn2NiGa3)
Mn2VAl1)
5.9370
5.9320
5.841 (1.6%)
5.818 (1.9%)
5.940 (0.1%)
5.832 (1.7%)
Mn2RuSn5)
1.68
1.888 (12%)
0.879 (47%)
Mn2NiGa3)
1.15
1.186 (3.1%)
0.792 (31%)
Mn2VAl1)
1.82
2.003 (10%)
1.971 (4.1%)
total DOS up
dn
−10
0
M n(D) tot up
dn
−10
M n(B) tot up
dn
Ru tot up
dn
−10
0
Si tot up
dn
0
0.5
−10
4.
Summary
We investigated a stable atomic arrangement and magnetic
ordering in Mn2RuSn and Mn2RuSi on the basis of firstprinciples total-energy calcu1ations. Our data show that the
most stable structure is a ferrimagnetic XA-type. This result
is reliable because X-ray diffraction measurements indicate
the material has a L21B- or a XA-type structure. For the XA,
the calculated values of lattice constants, magnetic moments
on constituent atoms and those per formula unit are in good
agreement with the experimental data. Our result indicates a
XA-type Mn2RuSi will be a half-metallic ferrirnagnet.
The authors would like to express sincere thanks to
Professor T. Kanomata at Tohoku Gakuin University for his
helpful discussions.
0
0.5
1
Then, we had better consider the effect of temperature. The
reason is as follows. There is a recent study12) of a
comparison between calculated and measured energies of
formation of Ni3Z (Z = Al, Ga, Ge) for two structures such
as L21A and AuCu3. Here, the calculated formation energy
¦E is defined as follows: ¦E = E(X2YZ) ¹ (2E(X) +
E(Y) + E(Z)). E(X2YZ) is a total energy of the material
X2YZ per formula unit (f.u.) and E(X) is a total energy of the
constituent atom X. The report shows that the measured
enthalpies of formation at a given T close to room temperature approximately equal the value of ¦E at zero Kelvin
with an error of 17 kJ/moll (0.18 eV/f.u.). In the present case
of Mn2NiGa, the XA-type structure has the lowest energy and
the L21A- and L21B-type structures have the second lowest
energy (the total-energy difference between L21A and L21B is
almost zero). The difference between the first and second
lowest energies is about 0.2 eV/f.u. This value is the same
order of the above mentioned error.
Acknowledgment
0
−10
315
Energy [eV]
0
Fig. 5 Density of states for Mn2RuSi. The top figure shows DOS per
formula unit and the others shows DOS of constituent atoms. The up-spin
(down-spin) state is depicted by a solid line (broken line). The Fermi level
is depicted by a vertical broken line at E = 0 eV.
shows that at the Fermi energy there exists DOS in an up-spin
state but does not exist in a down-spin state (see Fig. 5).
Finally, we make some comments on the result of
Mn2NiGa. Our conclusion is that if the compound holds
the parent phase without a martensite transformation, then it
has a ferrimagnetic XA-type structure as a ground state. In
fact, it undergoes a structural transformation near 500 K.
REFERENCES
1) K. H. J. Buschow and P. G. van Engen: J. Magn. Magn. Mater. 25
(1981) 90­96.
2) G. D. Liu, X. F. Dai, S. Y. Yu, Z. Y. Zhu, J. L. Chen and G. H. Wu:
Phys. Rev. B 74 (2006) 054435.
3) P. J. Brown, T. Kanomata, K. Neumann, K. U. Neumann, B. Ouladdiaf,
A. Sheikh and K. R. A. Ziebeck: J. Phys. Condens. Matter 22 (2010)
506001.
4) G. D. Liu, X. F. Dai, H. Y. Liu, J. L. Chen, Y. X. Li, G. Xiao and G. H.
Wu: Phys. Rev. B 77 (2008) 014424.
5) K. Endo, T. Kanomata, H. Nishihara and K. R. A. Ziebeck: J. Alloys
Compd. 510 (2012) 1­5.
6) P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka and J. Luitz:
WIEN2k, An Augmented Plane Wave + Local Orbitals Program for
Calculating Crystal Properties, (Karlheinz Schwarz, Techn. Universität
Wien, Austria, 2001).
7) J. P. Perdew, K. Burke and M. Emzerhof: Phys. Rev. Lett. 77 (1996)
3865­3868.
8) W. Kohn and N. Rostoker: Phys. Rev. 94 (1954) 1111­1120.
9) H. Akai and P. H. Dederichs: Phys. Rev. B 47 (1993) 8739­8747.
10) P. Soven: Phys. Rev. 156 (1967) 809­813.
11) X. Yan, A. Grytsiv, P. Rogl, V. Pomlakushin and M. Palm: J. Phase
Eguilib. Diffus. 29 (2008) 500­508.
12) M. Gilleßen and R. Dronskowski: J. Comput. Chem. 30 (2009) 1290­
1299.