Defects-driven appearance of half-metallic ferrimagnetism in Co-Mn--based Heusler alloys

Abstract

Half-metallic ferromagnetic full-Heusler alloys containing Co and Mn, having the formula Co$_2$MnZ where Z a sp element, are among the most studied Heusler alloys due to their stable ferromagnetism and the high Curie temperatures which they present. Using state-of-the-art electronic structure calculations we show that when Mn atoms migrate to sites occupied in the perfect alloys by Co, these Mn atoms have spin moments antiparallel to the other transition metal atoms. The ferrimagnetic compounds, which result from this procedure, keep the half-metallic character of the parent compounds and the large exchange-splitting of the Mn impurities atoms only marginally affects the width of the gap in the minority-spin band. The case of [Co$_{1-x}$Mn$_x$]$_2$MnSi is of particular interest since Mn$_3$Si is known to crystallize in the Heusler $L2_1$ lattice structure of Co$_2$MnZ compounds. Robust half-metallic ferrimagnets are highly desirable for realistic applications since they lead to smaller energy losses due to the lower external magnetic fields created with respect to their ferromagnetic counterparts.

Full text

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arXiv:cond-mat/0702437v1 [cond-mat.mtrl-sci] 19 Feb 2007
Defects-drivenappearanceofhalf-metallicferrimagnetisminCo-Mn–based
Heusleralloys
K.¨Ozdo˜ ganaI. Galanakisb,∗E. S ¸a¸ sıo˜ gluc,dB. Akta¸ sa
aDepartment of Physics, Gebze Institute of Technology, Gebze, 41400, Kocaeli, Turkey
bDepartment of Materials Science, School of Natural Sciences, University of Patras, GR-26504 Patra, Greece
cInstitut f¨ ur Festk¨ orperforschung, Forschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany
dFatih University, Physics Department, 34500, B¨ uy¨ uk¸ cekmece,˙Istanbul, Turkey
Abstract
Half-metallic ferromagnetic full-Heusler alloys containing Co and Mn, having the formula Co2MnZ where Z a sp element, are
among the most studied Heusler alloys due to their stable ferromagnetism and the high Curie temperatures which they present.
Using state-of-the-art electronic structure calculations we show that when Mn atoms migrate to sites occupied in the perfect
alloys by Co, these Mn atoms have spin moments antiparallel to the other transition metal atoms. The ferrimagnetic compounds,
which result from this procedure, keep the half-metallic character of the parent compounds and the large exchange-splitting of
the Mn impurities atoms only marginally affects the width of the gap in the minority-spin band. The case of [Co1−xMnx]2MnSi
is of particular interest since Mn3Si is known to crystallize in the Heusler L21 lattice structure of Co2MnZ compounds. Robust
half-metallic ferrimagnets are highly desirable for realistic applications since they lead to smaller energy losses due to the lower
external magnetic fields created with respect to their ferromagnetic counterparts.
Key words: A.magnetically ordered materials
PACS: 75.47.Np, 75.50.Cc, 75.30.Et
1. Introduction
The emergence of spintronics, also known as magneto-
electronics, the last decade brought in the center of scien-
tific research new properties and materials which had re-
ceived little attention in the past [1]. An important class of
materials which are at the present under intense study are
the so-called half-metals [2]. These materials are hybrids
between metals and semiconductors or insulators, present-
ing metallic behavior for one spin-band and semiconduct-
ing for the other, and thus overall they are either ferro-
or ferrimagnets with perfect spin-polarization at the Fermi
level [3]. de Groot and his collaborators in a pioneering
paper published in 1983 predicted the existence of half-
metallicity in the case of the intermetallic Heusler alloy
NiMnSb [4]. Since then several half-metallic ferromagnetic
∗Corresponding author. Phone +30-2610-969925, Fax +30-2610-
969368
Email addresses: kozdogan@gyte.edu.tr (K.¨Ozdo˜ gan),
galanakis@upatras.gr (I. Galanakis), e.sasioglu@fz-juelich.de
(E. S ¸a¸ sıo˜ glu).
materials have been initially predicted by theoretical ab-
initio calculations and after synthesized experimentally.
Half-metallic Heusler alloys are expected to play a key
role in realistic applications due to their very high Curie
temperatures and their structural similarity to the widely
used binary semiconductors crystallizing in the zinc-blende
structure [5]. Initially the research was focused on the so-
called half- or semi-Heusler compounds like NiMnSb [3,6]
but lately the interest has been shifted to the so called
full-Heusler compounds and mainly to the ones containing
Co, like Co2MnAl or Co2MnGe, which were already known
since early 70’s [7]. In early 90’s it was argued in two pa-
pers by a japanese group that they should be half-metallic
ferromagnets [8] and first-principles calculations by Picozzi
et al [9] and Galanakis et al [10] in 2002 confirmed their
predictions. These papers acted as an inspiration to ex-
perimentalists who devote a constantly increasing number
of publications to the study of their properties (see Refs.
[2,3,11,12,13,14,15,16,17,18,19,20] for references to some of
these studies). Both the origin of ferromagnetism [21] and
half-metallicity [10] in these compounds are well under-
stood.

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Although the research on half-metallic ferromagnets is
intense, the ideal case would be a half-metallic antiferro-
magnet, also known as fully-compensated ferrimagnet [22],
since such a compound would not give rise to stray flux and
thus would lead to smaller energy consumption in devices.
In the absence of such an ideal compound, half-metallic
ferrimagnetism is highly desirable since such compounds
would yield lower total spin moments than their ferromag-
netic counterparts. Some perfect Heusler compounds like
FeMnSb [23] and Mn2VAl [24] are predicted to present
half-metallicity in conjunction with ferrimagnetism. Re-
cently other routes to half-metallic ferrimagnetism have
been studied like the doping of diluted magnetic semicon-
ductors [25], and the inclusion of defects in Cr pnictides
[26] and Co2CrAl(or Si) full-Heusler alloys [27].
In this communication we will study the appearance of
defects-drivenferrimagnetisminthe popularCo2MnZcom-
pounds where Z stands for Al, Ga, Si, Ge or Sn. Most of the
theoretical studies on these compounds either are devoted
to the perfect compounds [10,28,29] or concern disorder be-
tween the Mn and the sp atoms or doping of the Mn sites
(seeRefs.[30,31,32]andreferencestherein).Whenasurplus
of Mn atoms is created occupying the sites corresponding
to Co atoms at the perfect compounds, [Co1−xMnx]2MnZ
alloys, these Mn impurity atoms are shown to be antifer-
romagnetically coupled to the other Co and Mn atoms at
the perfect lattice sites. The resulting ferrimagnetic com-
pounds keep the half-metallic character of the perfect par-
ent alloys.Interestinglydue to the veryhigh exchangesplit-
ting presented by the Mn impurity atoms, the width of the
gap is only marginally affected contrary to the Co2CrAl
and Co2CrSi where the creation of Cr antisites almost van-
ished the gap [27]. Thus the defects-driven half-metallic
ferrimagnetism presented in this communication is of par-
ticular interest for realistic applications. Special attention
should be given to [Co1−xMnx]2MnSi alloys since Mn3Si
is known experimentally to crystallize in the Heusler L21
lattice structure of Co2MnZ compounds with two equiva-
lent type of Mn atoms in the unit cell [33]. On the contrary
Mn3Ge and Mn3Sn compounds crystallize in an hexago-
nal structure [34] while no information is available on the
Mn3Al or Mn3Ga compounds. We should also note here
that Picozzi and collaborators studied in Ref. [35] the case
of asingleMn antisiteusing asupercellstructurein the case
of Co2MnSi and Co2MnGe compounds. They had actually
found that the Mn impurity atoms has a spin moment an-
tiparallel to the other transition metal atoms but they had
not considered the case of extensive defects or studied in
detail the observed behavior in their calculations.
We employed the full–potential nonorthogonal local–
orbital minimum–basis band structure scheme (FPLO)
to perform the electronic structure calculations and the
coherent potential approximation (CPA) to simulate the
creation of the Mn antisites [36]. We used the scalar rela-
tivistic formulation and thus the spin-orbit coupling was
not taken into account. The exchange–correlation poten-
tial was chosen in the local spin density approximation
-4-20
E (eV)
-4 -20
E (eV)
-5
0
5
-5
0
5
DOS (states/eV)
x=0.0
x=0.1
x=0.2
-5
0
5
[Co1-xMnx]2MnZ
Al Si
Ga Ge
Sn
Fig. 1.
alloys as a function of the concentration x : we denote x = 0 with
the solid grey line with shaded region, x = 0.1 with a solid thick blue
line and x = 0.2 with a dashed red line. The Fermi level has been
chosen as the zero of the energy axis, and positive values of DOS
correspond to the spin-up (majority) electrons while negative values
correspond to the spin-down (minority) electrons. In the insets we
have blown up the region around the Fermi level.
Total density of states (DOS) for the [Co1−xMnx]2MnZ
(LSDA). The self-consistent potentials were calculated on
a 20 × 20 × 20 k-mesh in the Brillouin zone, which corre-
sponds to 256 k points in the irreducible Brillouin zone.
The lattice constants were the experimental ones, 0.5756
nm for Co2MnAl, 0.577 nm for Co2MnGa, 0.565 nm for
Co2MnSi, 0.574 for Co2MnGe and 0.598 nm for Co2MnSn
[5], and were kept constant upon the creation of defects.
2. Results and discussion
Wehaveperformedcalculationsforthe[Co1−xMnx]2MnZ
compounds varying the sp atom, Z, which is one of Al,
Ga, Si, Ge or Sn. We have taken into account five different
values for the concentration x; x= 0, 0.025, 0.05, 0.1, 0.2.
In Fig. 1 we have drawn the total density of states (DOS)
for all five families of compounds under study and for
three different values of the concentration x : the perfect
compounds (x=0) and for two cases with defects, x= 0.1
and 0.2. In the case of the perfect Co2MnSi and Co2MnGe
compounds there is a real gap in the minority spin band
and the Fermi level falls within this gap and these com-
pounds are perfect half-metals. The other three perfect
compounds -Co2MnAl, Co2MnGa and Co2MnSn- present
in reality a region of tiny minority-spin DOS instead of a
real gap, but the spin-polarization at the Fermi level only
marginally deviates from the ideal 100% and these com-

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-4-20
E (eV)
-2
0
2
-2
0
2
DOS (states/eV)
x=0.0
x=0.1
x=0.2
-4-20
E (eV)
[Co1-xMnx]2MnAl
Co Mnimp
Mn 4 Al
x
-4-20
E (eV)
-2
0
2
-2
0
2
DOS (states/eV)
x=0.0
x=0.1
x=0.2
-4 -20
E (eV)
Co Mnimp
Mn 4 Si
x
[Co1-xMnx]2MnSi
Fig. 2. Atom-resolved DOS as a function of the concentration x for
the [Co1−xMnx]2MnAl (upper panel) and [Co1−xMnx]2MnSi (lower
panel) compounds as a function of the concentration x. We have
used the same notation as in Fig. 1. Note that the atomic DOS’s
have been scaled to one atom. The DOS for Al and Si has been
multiplied by a factor of four. Details are similar to Fig. 1.
pounds can be also considered as half-metals. These results
agree with previous electronic structure calculations on
these compounds [9,10,29,30,31] and are confirmed by the
calculated total spin moments as discussed latter in the
text. When we create a surplus of Mn atoms which migrate
at sites occupied by Co atoms in the perfect alloys, the
gap persists and all compounds retain their half-metallic
character. This is clearly seen in the insets where we have
blown the region around the Fermi level. The most inter-
esting case is the two compounds which presented a real
gap, Co2MnGe and mainly Co2MnSi. The creation of Mn
antisites, especially in the Si-case, does not alter the width
of the gap and half-metallicity is extremely robust in these
alloys with respect to the creation of Mn antisites. We
will explain this behavior latter in the text, when we will
discuss the atom-resolved DOS.
In Fig. 2 we have drawn the atom resolved DOS for the
Al- and Si-based compounds presented in Fig. 1. In the per-
fect compounds, x=0, the Co and Mn atoms form a com-
mon majority-spin band while the minority occupied states
aremainlyofCocharacter.Aroundtheminority-spinband-
gap the states are mainly of Co characterconfirming the re-
sults in Ref. [10]. The large exchange splitting between the
occupied majority-spin states of Mn atoms and its unoccu-
pied minority states favors the creation of the gap and the
appearanceof half-metallicity being alsoresponsiblefor the
large spin moments at the Mn sites (see next paragraph).
The DOS of the sp atoms is very small with respect to the
transition metal atoms and thus we have multiplied it by
four to make it visible. Al or Si states around the Fermi
level are of p character and they play a central role in the
exact position of the Fermi level within the gap (see Ref.
[3] for an extended discussion of the role of the sp atoms).
When we substitute Si for Al, we increase the total number
of valence electrons in the unit cell by one and this extra
electron does not occupy states of the sp atom, which are
deep in energy, but rather states of the transition metal
atoms [10] provoking small changes in the DOS of the Co
and Mn atoms. When we create Mn antisites, the DOS of
both Co and Mn atoms at the perfect sites does not sig-
nificantly change as can be easily observed in Fig. 2 and
thus they only marginally affect the half-metallicity. The
interesting phenomenon is the behavior of the Mn impuri-
ties atoms sitting at the antisites. Now the major weight
of the occupied states corresponds to spin-down states and
we expect their spin moments to be antiparallel to the ones
of the Co and Mn atoms at the perfect sites. Moreover the
large exchange splitting of the Mn atoms ensures a large
gap in the spin-down band and the robust character of the
half-metallicity. The shape of the DOS of the Mn impurity
atoms at the antisites is similar to the one of the Cr impu-
rity atoms in Co2CrAl and Co2CrSi [27]. The main advan-
tage of the Mn-based compounds is that when Cr substi-
tutes Mn, the smaller exchange splitting of the Cr atoms
leads to huge narrowing of the band-gap which shrinks to
an almost zero value [27]. Thus the Mn compounds have a
clear advantage for realistic applications.
Our discussion has been focused mainly to the half-
metallic character of the compounds with the Mn defects.
We will now concentrate on the spin moments of the
compounds under study to discuss the appearance of the
ferrimagnetism in the defected compounds. In Table 1 we
have gathered the atom-resolved and total spin moments
for the [Co1−xMnx]2MnZ alloy where Z is Al or its iso-
electronic Ga and for all values of the concentrations x
which we have studied, and in Table 2 we present the same
information for the Z= Si, Ge or Sn alloys. We will start
our discussion from the calculated total spin moments
and how they compare with the ideal values for perfect
half-metallicity (values in parenthesis in the tables). These
ideal values stem from the Slater-Pauling behavior shown
for the full-Heusler alloys [10] which states that the total
spin moment in the unit cell in µB is the total number
of valence electrons in the unit cell, zt, minus ”24” since
there are exactly 12 occupied minority-spin states for all
half-metallic full-Heuslers. For the perfect compounds this

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Table 1
Atom-resolved
[Co1−xMnx]2MnZ compounds (moments have been scaled to one
atom), where Z is Al or Ga. The last column is the total spin mo-
ment (Total) in the unit cell calculated as 2 × [(1 − x) ∗ mCo+ x ∗
mMn(imp)]+mMn+mZand in parenthesis the ideal total spin mo-
ment predicted by the Slater-Pauling rule for half-metals (see Ref.
[10]). With Mn(imp) we denote the Mn atoms sitting at perfect Co
sites.
spinmagneticmoments (in
µB) forthe
[Co1−xMnx]2MnAl
x
Co Mn(imp) Mn Al Total(Ideal)
0 0.68– 2.82 -0.14 4.04(4.0)
0.025 0.71-2.632.82 -0.14 3.92(3.9)
0.05 0.73 -2.59 2.82 -0.13 3.81(3.8)
0.10.78-2.49 2.83 -0.12 3.61(3.6)
0.20.84-2.23 2.85 -0.093.20(3.2)
[Co1−xMnx]2MnGa
x
Co Mn(imp) Mn Ga Total(Ideal)
0 0.65– 2.90 -0.104.09(4.0)
0.025 0.68-2.762.90 -0.104.00(3.9)
0.05 0.71-2.73 2.91 -0.093.89(3.8)
0.10.75-2.66 2.92 -0.08 3.66(3.6)
0.2 0.82-2.42 2.94 -0.05 3.23(3.2)
means a total spin moment of 4 µB for Co2MnAl and
Co2MnGa since they have 28 valence electrons in the unit
cell and a spin moment of 5 µBfor the Co2MnSi, Co2MnGe
and Co2MnSn compounds which have one electron more.
The calculated total spin moments are exactly 5 µBfor the
perfect Co2MnSi and Co2MnGe alloys and only slightly de-
viate from the ideal values for the other three compounds
confirming the discussion on the total DOS. When we in-
duce the Mn impurities at the antisites, we have to take
the average value of the valence electrons calculated as
2×[(1−x)∗zCo+x∗zMn(imp)]+zMn+zsp atomwhere z is
the number of valence electrons of the corresponding chem-
ical element. Mn is lighter than Co and thus as we increase
the concentration in Mn defects, the total spin moment
should decrease. In the case of the [Co1−xMnx]2Mn-Si
and -Ge compounds the ideal half-metallicity is preserved
irrespectively of the concentration of the Mn impurities.
The Al and Sn compounds are almost half-metallic while
slightly larger deviations are observed for the Ga com-
pound. Overall we could safely state that half-metallicity
is preserved upon the creation of Mn antisites.
The spin moment of Mn atoms at the perfect sites
remains practically constant for all five families of com-
pounds under study, when the concentration in Mn defects
is increased, since their environment does not signifi-
cantly change; each Mn atom has eight Co atoms as first
neighbors in the perfect alloy and for the values of the
concentration under study here their environment remains
of mainly Co character. Co atoms in the case of Si, Ge
and Sn compounds retain a practically constant moment
while in Al and Ga compounds it slightly increases. This
Table 2
Similar to Table 1 for the [Co1−xMnx]2MnZ compounds with Z=
Si, Ge or Sn.
[Co1−xMnx]2MnSi
x
Co Mn(imp) MnSi Total(Ideal)
0.00.98– 3.13 -0.095.00(5.0)
0.025 0.99-1.01 3.11 -0.094.90(4.9)
0.05 0.99-0.95 3.09 -0.084.80(4.8)
0.10.99-0.84 3.06 -0.074.60(4.6)
0.2 0.97 -0.702.99 -0.054.20(4.2)
[Co1−xMnx]2MnGe
x
Co Mn(imp) Mn Ge Total(Ideal)
0 0.93– 3.20 -0.065.00(5.0)
0.025 0.94-1.72 3.21 -0.064.90(4.9)
0.05 0.95-1.63 3.20 -0.054.80(4.8)
0.10.97-1.45 3.19 -0.044.60(4.6)
0.20.96-1.18 3.15 -0.024.20(4.2)
[Co1−xMnx]2MnSn
x
Co Mn(imp) Mn Sn Total(Ideal)
0 0.89– 3.32 -0.085.02(5.0)
0.025 0.91-2.44 3.34 -0.074.91(4.9)
0.05 0.93 -2.343.35 -0.064.81(4.8)
0.10.96-2.14 3.35 -0.054.60(4.6)
0.20.98-1.73 3.35 -0.034.20(4.2)
implies that in the case of Si, Ge and Sn-based compounds
Co atoms show a more atomic like behavior and they are
not strongly affected by their environment and thus their
spin moment is around 1 µB. In the case of Al and Ga
compounds Co atoms are more affected by their environ-
ment and for this reason their spin moment in the perfect
compounds is smaller. As we increase the concentration of
Mn defects in these two alloys, also the spin moment of
the Co atoms increases reaching a value of ∼ 0.8 µB. Since
the spin moments of both Co and Mn atoms at the per-
fect sites do not present drastic changes, the only way to
achieve the half-metallicity for the compounds (and thus
to decrease their total spin moment) is that the Mn impu-
rity atoms at the antisites have spin moments antiparallel
to the other transition metal atoms. This is actually the
case as can be seen in Tables 1 and 2. Mn impurity atoms
have an important negative spin moment especially in the
case of the Al and Ga compounds where it attends -∼
2.6-2.8 µB. It slightly drops as we increase the concentra-
tion in Mn defects but this is expected since we increase
also the hybridization between Mn impurity atoms; Mn
atoms contrary to Co atoms have more expanded d-like
wavefunctions and hybridize stronger between them.
Of particular interest is the case of Si , Ge and Sn -
compounds, where the spin moment of the Mn impurity
atoms shows large variations between the three families of
alloys. As we pass from Si to Ge and then to Sn-based al-

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loy, its spin moment for x=0.025 increases by -0.72 µB. Al-
though the spin moment of the Mn impurity atom changes
significantly, its contribution to the total spin moment is
small. A large increase of the absolute value of the spin
moment of the Mn impurities, as the one observed here,
means a small increase of its negative contribution to the
total spin moment which has to balance the larger positive
contribution of the spin moment of the Mn atoms at the
perfect sites observed for the compound with the heavier
sp element. For the Ge- and Sn-compounds the DOS has a
similar shape to the one for the Si-based alloys (and thus
they are not shown here), but the Fermi level is lower in
energy with respect the Si-based compound. Since the ma-
jority DOS takes large values at the region of the minority
gap, this small shift of the Fermi level provokes a consid-
erable increase of the negative spin moment of the Mn im-
purity atoms. Thus the Mn-doped alloys are half-metallic
ferrimagnets and their total spin moment is considerable
smaller than the perfect half-metallic ferromagnetic par-
ent compounds. A similar phenomenon has been also pre-
dicted when creating Cr antisites in the zinc-blende CrAs
intermetallic alloy [26] where Cr impurities spin moments
couple antiferromagnetically to the Cr atoms at the ideal
sites in the zinc-blende structure. Similarly Cr antisites in
Co2CrAl and Co2CrSi compounds lead to a similar phe-
nomenon [27]. The importance of the present results stem
from the more intense research on the compounds contain-
ing Co and Mn and from their more robust half-metallic
character,since in the caseof Cr compounds the gapalmost
vanishes upon the creation of Cr defects. Here we have to
mention that if also Co atoms migrate to Mn sites (case
of atomic swaps) the half-metallity is lost, as it was shown
by Picozzi et al. [35], due to the energy position of the Co
states which have migrated at Co sites.
3. Summary and conclusions
We have studied the effect of defects-driven appearance
of half-metallic ferrimagnetism in the case of the Co2MnZ
Heusler alloys, where Z stands for Al, Ga, Si, Ge or Sn.
More precisely, based on first-principles calculations we
have shown that when we create Mn antisites at the Co
sites, these impurity Mn atoms couple antiferromagneti-
cally with the Co and the Mn atoms at the perfect sites
while keepingthe half-metallic characterof the parent com-
pounds. Half-metallicity in these compounds is a robust
property since the large exchange splitting of the Mn impu-
rity atoms ensures that the width of the gap in the minority
spin band only marginally is affected. Thus we have shown
an alternative way to create robust half-metallic ferrimag-
nets, which are crucial for magnetoelectronic applications,
based in the introduction of defects in half-metallic ferro-
magnetic Heusler alloys which are widely studied. Espe-
cially the case of [Co1−xMnx]2MnSi alloys is of particular
interest since Mn3Si is known to crystallize in the Heusler
L21lattice structure of Co2MnSi. We expect these results
to stimulate further interest in the theoretical and experi-
mental research in the field of spintronics.
References
[1] I.ˇZuti´ c, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76 (2004)
323.
[2] Half-metallic alloys: fundamentals and applications, Eds.: I.
Galanakis and P.H. Dederichs, Lecture notes in Physics vol. 676
(Berlin Heidelberg: Springer 2005).
[3] I. Galanakis, Ph. Mavropoulos, P.H. Dederichs, J. Phys. D: Appl.
Phys. 39 (2006) 765.
[4] R.A. de Groot. F.M. Mueller, P.G. van Engen, K.H.J. Buschow,
Phys. Rev. Lett. 50 (1983) 2024.
[5] P. J. Webster and K. R. A. Ziebeck, in Alloys and Compounds
of d-Elements with Main Group Elements. Part 2., edited by
H. R. J. Wijn, Landolt-Bo¨ ornstein, New Series, Group III, Vol.
19,Pt.c (Springer-Verlag, Berlin 1988), pp. 75-184.
[6] I. Galanakis, P.H. Dederichs, N. Papanikolaou, Phys. Rev. B 66
(2002) 134428.
[7] P.J. Webster, J. Phys. Chem. Solids 32 (1971) 1221.
[8] S. Ishida, S. Fujii, S. Kashiwagi, S. Asano, J. Phys. Soc. Jpn. 64
(1995) 2152; S. Fujii, S. Sugimura, S. Ishida, S. Asano, J. Phys.:
Condens. Matter 2 (1990) 8583.
[9] S. Picozzi, A. Continenza, A.J. Freeman, Phys. Rev. B (2002)
094421.
[10] I. Galanakis, P.H. Dederichs, N. Papanikolaou, Phys. Rev. B 66
(2002) 174429.
[11] A. Bergmann, J. Grabis, B.P. Toperverg, V. Leiner, M. Wolff, H.
Zabel, K. Westerholt, Phys. Rev. B 72 (2005) 214403; J. Grabis,
A. Bergmann, A. Nefedov, K. Westerholt, H. Zabel, Phys. Rev.
B 72 (2005) 024437; idem, Phys. Rev. B 72 (2005) 024438.
[12] A. Rata, H. Braak, D.E. B¨ urgler, S. Cramm, and C.M. Schneider,
Eur. Phys. J. B 52 (2006) 445; M. Kallmayer H. Schneider,
G. Jakob, H.J. Elmers, K. Kroth, H.C. Kandpal, U. Stumm,
S. Cramm, Appl. Phys. Lett. 88 (2006) 072506; S.V. Karthik,
A. Rajanikanth, Y.K. Takahashi, T. Okhubo, K. Hono, Appl.
Phys. Lett. 89 (2006) 052505; G.H. Fecher, H.C. Kandpal, S.
W¨ urmehl, J. Morais, H.-J. Lin, H.-J. Elmers, G. Sch¨ onhense, C.
Felser, J. Phys.: Condens. Matter 17 (2005) 7237; R.Y. Umetsu,
K. Kobayashi, A. Fujita, K. Oikawa, R. Kainuma, K. Ishida,
N. Endo, K. Fukamichi, A. Sakuma, Phys. Rev. B 72 (2005)
214412; K. Kobayashi, R.Y. Umetsu, R. Kainuma, K. Ishida, T.
Oyamada, A. Fujita, K. Fukamichi, Appl. Phys. Lett. 85 (2004)
4684; H.J. Elmers, G.H. Fecher, D. Valdaitsev, S.A. Nepijko, A,
Gloskovskii, G. Jakob, G. Sch¨ onhense, S. Wurmehl, T. Block,
C. Felser, P.-C. Hsu, W.-L. Tsai, S. Cramm, Phys. Rev. B 67
(2003) 104412.
[13] T. Marukame, T. Ishikawa, K.-I. Matsuda, T. Uemura, M.
Yamamoto, Appl. Phys. Lett. 88 (2006) 262503; T. Marukame,
T. Kasahara, K. Matsuda, T. Uemura T M. Yamamoto, Jpn. J.
Appl. Phys. 44 (2005) L521.
[14] R. Kelekar and B.M. Klemens, Appl. Phys. Lett. 86 (2005)
232501; R. Kelekar, H. Ohldag, B.M. Clemens, Phys. Rev. B 75
(2007) 014429.
[15] S. K¨ ammerer, A. Thomas, A. H¨ utten, G. Reiss, Appl. Phys.
Lett. 85 (2004) 79; J. Schmalhorst, S. K¨ ammerer, M. Sacher, G.
Reiss, A. H¨ utten, A. Scholl, Phys. Rev. B 70 (2004) 024426.
[16] Y. Sakuraba, M. Hattori, M. Oogane, Y. Ando, H. Kato, A.
Sakuma, T. Miyazaki, H. Kubota, Appl. Phys. Lett. 88 (2006)
192508; Y. Sakuraba, J. Nakata, M. Oogane, Y. Ando, H. Kato,
A. Sakuma, T. Miyazaki, H. Kubota, Appl. Phys. Lett. 88
(2006)022503; Y. Sakuraba, T. Miyakoshi, M. Oogane, Y. Ando,
A. Sakuma, T. Miyazaki, H. Kubota, Appl. Phys. Lett. 89 (2006)
052508; X.Y. Dong, C. Adelmann, J.Q. Xie, C.J. Palmstr¨ om, X.