J. Magn. Soc. Jpn., 41, 81-93 (2017)<Review>

Basic and Applied Research of Transition-metal Based Magnetic Materials
K. Fukamichi
Professor Emeritus, Tohoku University, Sendai, Japan

Various transition-metal magnetic materials have been investigated from basic and practical viewpoints. The
concentration dependence of the Néel temperature TN of Cr-based alloys is complicated. Cr-Si and Cr-Fe
antiferromagnetic alloys show Invar characteristics in the ternary alloys. Fe-based amorphous alloys exhibit weakferromagnetic properties, resulting in remarkable magnetovolume effects.
The icosahedral quasicrystals
containing Mn show a spin-glass behavior, in a similar manner as those of amorphous counterparts.
Itinerant-electron metamagnetic transition occurs in La(FexSi1-x)13, accompanied by many drastic change inmagnetic and elastic properties. These drastic changes are practically useful in the field magnetic refrigeration andlinear magnetostriction. The magnitude of TN of Mn-based -phase is increased by addition of Ir, Ru, Rh and thespin structures change, depending on temperature and composition. Several kinds of L10-type Mn alloys have ahigh value of TN with a large magnetocrystalline anisotropy. The shift of the exchange-bias field for the collinear
spin structure in L10-type phase is induced by spin frustration. L2 1- and B2-type Co2CrGa metallurgical stablealloys exhibit a high spin polarization. Several kinds of L21-type and B2-type alloys show a large ferromagnetic
shape memory effect associated with twin-boundary motions.
Key words: antiferromagnetic Invar alloy, weak ferromagnetic Fe-based amorphous alloy, quasicrystalline alloy,
spin-glass, itinerant-electron metamagnetic transition, magnetic refrigerant, giant magnetostriction, exchange- bias
field, half-metalicity, ferromagnetic shape memory alloy
1. Introduction
In the present article, I introduce my several
research subjects for the MSJ Award 2016. I haveengaged in basic and applied research for a variety oftransition-metal magnetic materials during several
decades.
Listed below are our main subjects:
antiferromagnetism and Invar characteristics of
Cr-basedalloys,weak ferromagnetic
Fe-based
amorphous alloys, Hall resistivity of amorphous alloys,
comparison
betweenmagnetic
properties
inquasicrystalline and amorphous states of Al-based Mn
alloys, perpendicular magnetic anisotropy of Tb–Fe bulk
amorphous alloys, random magnetic anisotropy andcoercive field of Fe–rare earth amorphous alloys, giant
magnetoresistance effect in Cu–Co granular alloys,
linear magnetostriction of Fe– Pd and Fe–Ga alloys,

Fig.1 Correlation chart between several kinds ofresearch topics in magnetism and magnetic
materials.

itinerant-electron metamagnetism of Laves phasecompounds and NaZn13-type compounds exhibiting large
magnetocaloric effects (MCE), spin fluctuations in -Mn
alloys, antiferromagnetism, magnetocrystalline anisotropy and exchange coupling in Mn alloys,
magnetocrystalline anisotropy in Fe–Pd and Fe–Pt
alloys, half-metalicity and magnetic-field induced shapememory effect in Heusler type alloys.
These
achievements have been contributed to publish several
kinds of international monographs 1-8). I present theresults for the selected topics given in Fig. 1. The
neighboring research fields in the figure share technical,
magnetic and physical common terms, providing
research continuity.
2. Explanation of main topics
2.1 Antiferromagnetism and Invar characteristics of
Cr-based alloys.
Cr with a bcc structure orders antiferromagneticallyin a spin-density-wave (SDW) structure below TN = 311
K, showing a prototypical itinerant antiferromagnet. It
has been pointed out that pure Cr indicates a weak
first-order transition at TN. Cr-based alloys exhibit
three distinct ordered phases transverse SDW (AF1),
longitudinal SDW (AF2), and commensurate SDW (AF0).
The magnetic phase diagrams for Cr-based alloys havebeen studied as functions of concentration, pressure,temperature, and magnetic field 9).
We investigated magnetic properties of a number ofCr-based alloys. The transition metals at the right of
Cr in the periodic table, except for ferromagnetic

Journal of the Magnetics Society of Japan Vol.41, No.5, 201781

𝑸𝑸± =

2𝜋𝜋(1 ± 𝛿𝛿),𝑎𝑎 

thermal expansion is associated with the cancelation of
the phonon part of thermal expansion by the
spontaneous volume magnetostriction. Reducing thesteep volume change by addition of the third element to
Cr-Fe and Cr-Si binary alloys, Invar characteristics was
obtained below TN. Practically, we require to set Invar
characteristics in the vicinity of room temperature. Bychanging the electron concentration, the value of TN istunable. From these data, we can make suitablenon-ferromagnetic (≡ antiferromagnetic) Invar alloys.
Figure 2 shows the thermal expansion curves of
Cr–Fe–Mn alloys15). For comparison, we present thethermal expansion curve of a conventional Invar alloyLinear thermal expansion, L/L (K)elements Fe, Co and Ni, and paramagnetic Pd, increase
TN. While the transition metals at the left of Cr in the
periodic table decrease TN. For non-transition metal
additions, such as Al (above 2%), Ga (above 1%), Ge, Sn,
As and Sb increase TN, although Si decreases it10).
The wave vector, Q, of the SDW is along one of the
cube axes and is incommensurate with the periodicity of
the lattice. The wavelength of the SDW is about 25
unit cells at room temperature and is temperaturedependent. The 100 cross sections of the Fermi
surface centered at  and H are the inter-sections of theoctahedrons; the former is the electron octahedron andthe latter the hole octahedron. The antiferromagnetic
state is stabilized by the nesting due to the coulombinteraction between the  and H surfaces. When the
SDW is shifted in the direction of <100>the wave vector
Q is given as(1)

where a is the lattice constant and  is the nesting
parameter11,12). The value of  increases with theintroduction of the transition metals at the left of Cr in
the periodic table, as for V and Ti, for example.
Meanwhile,  decreases and vanishes for certain
concentration of metals, as with the introduction of
3d-metal impurities like Mn, Fe, Re, changing SDW to
AF0, i.e.,  = 0.
The Néel temperature TN, which is an experimental
measure of stability of the SDW, is given by an equationof the form
1
𝑇𝑇N = 𝑇𝑇0 exp (− ),𝜆𝜆(2)𝛾𝛾 2 𝑉𝑉(0)(𝑘𝑘c )2.
2𝜋𝜋 2 𝑣𝑣

(3)

where To is a function which depends on the band structure and𝜆𝜆 =Here  is the mean overlap matrix element for electrons
in the same band, V (0) the average screened Coulomb
potential, kc the wave vector in the first Brillouin zoneand  is the arithmetic mean of the Fermi velocities in
two bands. It is considered to be senseless to predict TNfrom Eq. (2) because of the exponential dependence onsome of the quantities involved11). In such complicatedcircumstances, it is considered to be practically usefulthat Cr–Fe13) and Cr–Si14) alloys exhibit a strong
first-order magnetic phase transition, accompanied by alarge spontaneous volume magnetostriction below TN in
the AF0 structure.
Before LCD (liquid crystal display), CRT (cathoderay tube display) had been used for long time. The CRT
has an Invar alloy as a shadow mask. However,conventional Invar alloys are ferromagnetic, and thenremanent magnetization and geomagnetism involve
color drifts offensive to the eye. Therefore, developmentof non-ferromagnetic Invar alloys was a pressing issue atthat time. The Invar characteristics, that is, a very low

82

Fe-36 wt%Ni
Invar

2  10-4

TN

TN

(a)
TN

(b)

TN

(c)
(d)

190

230

270

310

350

390

430

Temperature, T (K)

Fig. 2 Thermal expansion curves of Cr–Fe–Mn
antiferromagnetic alloys, together with the curveof conventional Invar alloy (Fe–36 wt% Ni)15).
(a) Cr-5.5%Fe-0.5%Mn, (b) Cr-25.5%Fe-1.0%Mn,(c) Cr-4.3%Fe-0.5%Mn, (d) Cr-4.2%Fe-1.0%Mn.
(Fe-36%wt%Ni). It is clear that the curves of Cr–Fe–Mn alloys exhibit an excellent Invar characteristics.
On the basis of data accumulation, we developed manykinds of Cr–Fe and Cr–Si based antiferromagnetic
Invar-type ternary alloys 10,16,17) . Figure 3 lists thepossible combinations of elements, which are selectedfrom the right upper grey area. In the figure, thedot-dashed line divides the border above which the Néel
temperature TN of Cr increases and below which it

Mn, Ru, Rh As, Sn
Re, Ir, Pt, Ge, Sb
Cr
Fe, SiCo,Pd,
Pd, Ni, V,
Co,
Mo, W, Ti, Nb

Fig.3

TN

Combination of elements for Cr-based
antiferromagnetic Invar-type ternary alloys 10,16,17).

Journal of the Magnetics Society of Japan Vol.41, No.5, 2017

decreases. What should be noted is that Co and Pd
decrease TN of Cr, yet increase TN of Cr–Fe and Cr–Si
alloys, exhibiting Invar characteristics around room
temperature1,10).
2.2. Weak ferromagnetic Fe-based amorphous alloys
2.2.1 Large magnetovolume and magnetoelastic
properties of Fe-based amorphous alloys
Amorphous magnetic alloys provide many peculiar
properties because of loss of lattice periodicity. Various
amorphous alloys have been prepared by melt-quenchingin a ribbon state and sputtering in a film state.
Fe–metalloid systems in the equilibrium phase diagram
have a relatively low eutectic point, showing an
excellent forming-ability of amorphous ribbons. The
striking aspect is that the number of nearest-neighboratom in transition metal–metalloid systems is estimatedto be about 13, being very close to that of the fcc lattice.
Concentration dependence of the magnetic moment per
Fe atom for amorphous Fe‒B alloys is not monotonic anda maximum occurs around 14 at% boron18), but then the
Curie temperature TC decreases with a decrease in theboron content. Such peculiar concentration dependences are similar to those of Fe–Ni fcc alloys which areclassified as weak ferromagnets with an incomplete
filling of up-spin band, resulting in Invar effects such asa low thermal expansion and a strong pressure effect on
the Curie temperature TC.
On thermal expansion curves of amorphous Fe‒B andFe‒P alloys, a large anomaly due to a remarkable
spontaneous volume magnetostriction is observed in awide temperature range below the Curie temperatureTC18). For Fe83B17 alloy, the value of the spontaneousvolume magnetostriction s estimated from the thermal
expansion curve is about 1x10-2, being of the same ofmagnitude as that of crystalline Fe‒Ni Invar alloys. In
addition, the pressure coefficient of the Curie
temperature, TC / P, is very large negative; the largest
negative value is of about –7 K/kbar, corresponding tothat of Fe71Ni29 crystalline fcc alloy. According toWohlfarth’s
theoryonvery
weak
itinerant
ferromagnetism19), the pressure effect on TC is given by 𝑇𝑇𝐶𝐶
𝐴𝐴
=− ,
 𝑃𝑃
𝑇𝑇𝐶𝐶

(4)

where the parameter A is proportional to the density ofstates, compressibility, effective degeneracy temperatureand the Bohr magneton. The data of Fe–Ni crystalline
Invar alloys are in line with A = 2050 K2/ kbar, and theseof Fe–B amorphous alloys fit approximately with A =3500 K2/kbar. The compressibility  calculated from theshear modulus and Young’s modulus for Fe–B alloys islarger than that of Fe‒Ni crystalline Invar alloys. Thatis, the value of the former is larger than that of the lattervalue by a factor of about 1.720).
This result isqualitatively consistent with Wohlfarth’s theory because
the value of A in Eq. (4) is proportional to .

Hitherto we have demonstrated that the magnetic
properties of Fe‒B amorphous Invar alloys are very
similar to those of Fe‒Ni crystalline Invar alloys. It
has been well that Invar behavior of Fe‒Ni crystalline
alloys is very complicated at lower Ni concentrations
because of the occurrence of a martensitic
transformation.
On the other hand, the magnetic
properties of amorphous alloys are studied without
taking such a complicated effect into consideration,
because amorphous alloys involve no martensitic
transformation. Accordingly, these results exclude a
conventional explanation of the origin of Invar effects
associated with the pre-martensitic transformation.
It is notable point that Fe–B amorphous alloys also
exhibit a remarkably large E effect defined as E /E 0with E =E s–E o, where E s and E 0 are Young’s moduli
in saturating field and in zero magnetic field,
respectively.
This effect is responsible for the Elinvar
characteristics21). The temperature coefficient of thedelay time t of delay lines is expressed as1
𝑡𝑡 = − (𝛼𝛼 + 𝑒𝑒),      
2(5)

where  and e are the temperature coefficients ofthermal expansion and Young’s modulus, respectively.
Fe‒B amorphous alloys indicate a very small
temperature coefficient of the delay time, because they
have the Invar and Elinvar characteristics at the same
temperature, in contrast to Fe–Ni crystalline Invar
alloys which exhibit a large value of e at room
temperature.
2.2.2 Forming-ability and weak ferromagnetism andspin-glass behavior of E–Fe alloy systems
There are several kinds of electron theories forforming-ability of amorphous films.
According toMoruzzi et al. theory22), a high Fermi level state density
means a lack of structural stability, resulting in an
amorphous state. In the case of early transition metal(E)–Fe systems, the larger the atomic size difference
becomes, the higher the Fermi level becomes23). We
systematically investigated the lower limit of solute
elements vs. the size difference ratio between Fe and thesolute clement. It was pointed out that the lower limit
of forming-ability of amorphous films depends not only
on the size difference ratio but also on the valencedifference between E and Fe3,24). That is, the lower
limit shifts to lower concentration ranges withincreasing values of the size and valence differences.
The energy separation in the density of state is roughly
proportional to the valence difference and leads to thegradual formation of a well-defined gap for compounds
22); namely, split-band state densities are formed byconstituents with large valence differences, leading to
the stabilization of an amorphous phase. Therefore, theeffect of the size difference on the forming-ability is
explained by the theory proposed by Moruzzi et al.
In Friedel's model25) which is valid in the strong

Journal of the Magnetics Society of Japan Vol.41, No.5, 201783

ferromagnetic situation, the charge displaced by theearly transition metal atom is related to themagnetization change by the simple formula 10–z,
where z is the host-solute valence difference. If theassumptions underlying this simple relation wereobeyed over the entire concentration range, then theconcentration dependence of the average moment ̅ isdefined as̅ = 0 − 𝑥𝑥(10 − 𝑧𝑧),

(6)

Average atomic moment,  (B)

where x is the atom fraction and 0 is the host momentat x=0. Friedel's formula Eq. (6) follows from theassumption that the repulsive solute potential displaces
precisely five majority-spin states from below to above
the Fermi level EF. Implicit in the Friedel formula,
therefore, is the assumption that the state density
immediately above the d-bands of the paramagnetic hostis negligible; this assumption is a necessary condition forstrong magnetism. Figure 4 displays the concentration
dependence of the mean magnetic moment ̅ of ExFe1-x
with E = Lu, Y, Th, Hf, Zr, Ti, Ta, Nb, V 26). The value of̅ becomes larger as the differences of the atomic sizeand the valence between E and Fe increases. Such asystematic tendency is explained from the bandcalculation23). In the figure, three dotted straight lines
indicate the expected line from Friedel's model which isvalid for strong ferromagnets. Since the observed
values are smaller than the expected values of dottedstraight lines, ExFe1-x amorphous systems are regarded
as weak ferromagnets which have holes in both d-bands.

2.4

3(Y, Lu)4(Th, Hf, Zr, Ti)
5(Ta, Nb, V)

2.0
1.6
1.2
0.8
0.4
0.0
0.0

Fe

㻌㻌Lu
Y
◇
Hf
◆
Zr
▼
Ta
●
Th
▲
Ti
＋
V
□
Nb
○
■

ExFe1-x
0.1
0.2
0.3
0.4
Atom fraction x solute E

0.5

0.6

Fig.4 Average low temperature magnetic moment
̅ of amorphous Fe alloys with many kinds of earlytransition metal solutes (E) as a function of atomic
fraction x26).
The Curie temperature TC of ExFe1-x amorphous films
with E=La, Lu, Ce, Hf and Zr exhibits a broad maximum
at x=0.2 and the ferromagnetic state diapers aroundx=0.1, resulting in a spin-glass state below about 110
K3,27).
This spin-glass state is caused by spin
frustrations in the magnetically concentrated ExFe1-x

84

amorphous alloys. It is meaningful to note that thenearest neighbor distance of Fe, d Fe-Fe, is about 2.5 Å28),very close to the critical distance of ferromagnetic andantiferromagnetic interactions.
2.3 Comparison between magnetic properties inquasicrystalline and amorphous states of Al-based Mn
alloysElectron diffraction studies revealed that Al–Mn
quenched alloys exhibit sharp spots with five-foldsymmetry axes29). This discovery contradicts a well-known crystallographic theorem which states that
five-fold symmetry axes can never appear in substanceshaving a truly long-range periodicity. A quasiperiodic
crystal, or quasicrystal, is a structure that is ordered but
not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational
symmetry. While crystals, according to the classicalcrystallographic restriction theorem, can possess only
two, three, four, and six-fold rotational symmetries, theBragg diffraction pattern of quasicrystals shows sharp
peaks with other symmetry orders, for instance five-fold.
It has been pointed out that computer simulations of
the supercooled Lennard Jones pair potential suggest
the existence of bond orientation order in liquids 30).
Because icosahedral clusters which are composed of 13
atoms have a significantly lower energy compared withthat of nuclei of fcc and hcp crystals, icosahedra should
be prevalent in liquids near the melting temperature
31,32).
It has been remarked that an amorphous
Lennard-Jones packing has about 32% of its atoms on
icosahedral sites33), and the scattering patterns observedin several amorphous alloys are quite similar to thosefrom icosahedral clusters. The sign of the exchangeinteraction oscillates between positive and negative
depending on the Mn‒Mn distance. It is therefore
interesting to study the magnetic properties ofquasicrystalline and amorphous alloys containing Mn.
We investigated magnetic properties of Al–Cu–TM
and Al–Pd–TM (TM: transition metal) quasicrystalline
alloys4,34). The difference between the quasicrystalline
alloys and their amorphous counterparts for magnetic
properties was confirmed in Al–Cu–Mn and Al–Pd–Mn
alloy systems in contrast to Al–Mn alloy system.
The magnetization is much smaller than the value
expected from the effective magnetic moment Peff (B),indicating that the magnitude of magnetic moment is not
uniform, depending on the Mn sites in thequasicrystalline phase.
From many kinds ofexperiments, it was confirmed that there are magnetic
and non-magnetic Mn atoms. The Curie constant C iswritten as
𝐶𝐶 =𝑁𝑁𝑁𝑁B 2 < 𝑃𝑃c > (< 𝑃𝑃c > +2)
,
3𝑘𝑘B

(7)

where N is the atomic number35). The average local

Journal of the Magnetics Society of Japan Vol.41, No.5, 2017

Table 1 The ratio of magnetic Mn atoms, X m/X,
the effective magnetic moment Peff, the spin
freezing temperature T f for Al85Mn15, Al65Cu20Mn15
and Al65Pd20Mn15 alloys in icosahedral quasicrystalline (Q) and amorphous (A) states34,36).
X m/X (%)

Peff (B)

T f (K)

Q-Al85Mn15

9*

0.59

–

A-Al85Mn15

9*

0.59

–

Q-Al65Cu20Mn15

8.9

1.56

6.1

A- Al65Cu20Mn15

18.7

2.24

8.1

Q-Al65Pd20Mn15

9.3

2.25

10.0

A- Al65Pd20Mn15

18.7

2.54

13.9

Alloy

2.4.1 Giant magnetocaloric and magnetostriction of
La(FexSi1-x)13 system
The itinerant-electron metamagnetic (IEM) transition is the field-induced first-order transition from theparamagnetic (PM) to the ferromagnetic (FM) state in
the itinerant-electron system. Wohlfarth and Rhodes37)
first discussed it on a phenomenological Landau theory.
The relation between the magnetic free energy F (M) andthe magnetization M is given as1
1
1
𝐹𝐹(𝑀𝑀) = 𝑎𝑎𝑀𝑀2 + 𝑏𝑏𝑀𝑀4 + 𝑐𝑐𝑀𝑀6 ,2
4
6*: extrapolated from the high Mn concentration range
magnetic moment per Mn atom, Pc, which corresponds
to the saturation magnetization is deduced from theCurie constant by assuming g factor to be 2. Using thevalues of the Curie constant, C, and the saturation
magnetization per Mn atom, M s, one can evaluate theatomicfraction,
Xm
from
the
following
relations35):
𝑋𝑋
3𝑘𝑘B 𝐶𝐶𝑀𝑀s ( 𝑀𝑀s + 2𝜇𝜇B ) =
.
𝑁𝑁
𝑋𝑋m

2.4 Itinerant-electron metamagnetism of NaZn13-type
compounds(8)

Comparing with the same magnitude of Peff (B), thecomposition of Mn for Al–Mn is much higher than thatfor the other two alloy systems, suggesting that theformation of the localized magnetic moment in Al–Mn ismuch difficult than in the latter systems. The value of
Peff (B) of Al–Mn in the amorphous (A) andquasicrystalline (Q) states is the same each other.
However, the difference between the amorphous (A) andquasicrystalline (Q) states for Al–Cu–Mn and Al–Pd–Mn
alloy systems is distinct as seen from Table 134). Notethat the latter two alloy systems contain Cu or Pd in
which the localized magnetic moment is easily
established. In Table 1, the ratios of magnetic Mn
atoms X m/X in Al–Cu–Mn and Al–Pd–Mn quasicrystalline alloys are about one half that of theamorphous counterparts, although there is no essential
distinction in Al–Mn alloy systems34,36). The spin-glass
behavior has been confirmed in both Al–Cu–Mn and
Al–Pd–Mn alloy systems even below 15 %Mn, although
Al–Mn alloys exhibit the spin-glass behavior above
20 %Mn. In addition, it is a noteworthy fact that
Al70Pd15Mn15 quasicrystalline alloy has a giant magnetic
moment in a similar manner as Pd–Mn crystalline dilutealloys. The difference between the A- and Q-state of thespin-glass freezing temperature T f, and the ratio of
magnetic Mn atom as well, arises from the looser
packing structure of the A-state, resulting in a larger
Mn–Mn distance.

(9)

where the Landau expansion coefficients a, b and c are
related to the 3d-electron band structures at the Fermi
level EF. The conditions of a > 0, b < 0, and c > 0 with
3/16 < a c /b2 < 9/20 are essential for the metamagnetictransition. The IEM transition is related to the onset ofthe exchange splitting in the band structure by applying
magnetic field.
This metamagnetic transition istermed ‛Itinerant-electron metamagnetism (IEM)’,different from a conventional metamagnetism in thelocalized electron system, where the metamagnetictransition (MT) takes place from the AFM to the FM
state. We experimentally confirmed for the first timeby using pulsed ultra-high magnetic fields that thevalues of the critical field for the metamagnetictransition of exchange-enhanced para-magnets YCo2
and LuCo2 Laves phase compounds are 69 and 71 T,
respectively38). These values are relatively smaller
than the theoretical values, which is reasonably
explained by taking into magnetovolume effect.
Furthermore, the IEM transition phase diagrams ofquasi-binary systems of Y(Co1-xMx)2 and Lu(Co1-xMx)2 (M:Al, Ga, Si)39) are consistent with the theoretical phase
diagram obtained by taking into the effect of spin
fluctuations40).
La(FexAl1−x)13 compounds crystallize in the cubic
NaZn13-type structure composed of icosahedral clusters
referred in Section 2.3 with two crystallographic sites: 8b
and 96i. The 8b site at the center of the icosahedralcluster is occupied by FeI atom only. The 96i sites at
the apex of the icosahedron are randomly occupied by
FeII atoms and Al atoms41). In these compounds, thehigh coordination numbers of Fe atoms and the shortest
Fe–Fe interatomic distance of d Fe-Fe 2.5Å are realized.
Such a densely packed structure with Fe atoms results
in a variety of magnetic states with a different volume
related to the itinerancy of Fe3d-electron42). Note thatthe icosahedra exist even in La(FexAl1−x)13 in theamorphous state5). Furthermore, we demonstrated that
La(FexSi1−x)13 compounds exhibit the IEM transition43).
The IEM transition is accompanied by drastic changes inmagnetic and elastic properties due to the first-orderphase transition. These changes are very attractivefrom the practical point of view. The cubic NaZn13-type

Journal of the Magnetics Society of Japan Vol.41, No.5, 2017

85

∂𝑀𝑀
∂𝑆𝑆
( ) =( ) ,
∂𝑇𝑇 H
∂𝐻𝐻 T

(10)

and hence the isothermal magnetic entropy change S m
is expressed as
H∂𝑀𝑀∆𝑆𝑆m = ∫ ( ) d𝐻𝐻.
∂𝑇𝑇 H
0

(11)

From Eq. (11), it is expected that the magnitude of Smbecomes very large when the magnetic transition is offirst-order. Be aware that the magnetization change at
the first-order transition is ideally discontinuous, andthen the value of ∂M / ∂T does not exist. Under such a
condition, S m should be calculated by theClausiusClapeyron equation given by the following
equation:∆𝑇𝑇∆𝑀𝑀|| = | |,
∆𝐻𝐻c
∆𝑆𝑆

(12)

where ΔM is the difference between magnetization
before and after the discontinuity for a given T, ΔHC is
the shift of critical field from ΔT and ΔS is the difference
between the entropies of the two phases.
In
experiments, however, a complete discontinuous
magnetization data at the first-order transition
temperature is scarcely obtained due to frozen
disorders45).
Figure 5 illustrates the temperaturedependence of S m in the magnetic field change from 0
to 5 T (0H =5 T) obtained by using Eq. (11) for
La(Fe0.88Si0.12)13 compound44).
The value of M / T
exhibits a large value around TC because of thethermal-induced first-order transition, and TC is
increased by the magnetic field, keeping a large value ofM / T. Consequently, S m in 0H =5 T indicates anegative maximum value of 22.3 J/kgK at TC =195 K.
Note that the magnitude of Sm obtained from Eq.(11) iscomparable with that calculated from Eq. (12). The
adiabatic temperature change Tad is obtained from𝐻𝐻

𝑇𝑇 𝜕𝜕𝜕𝜕( ) 𝑑𝑑𝑑𝑑
0 𝐶𝐶H 𝜕𝜕𝜕𝜕 H∆𝑇𝑇ad = − ∫= [𝑇𝑇(𝑆𝑆)H − 𝑇𝑇(𝑆𝑆)0 ]S ≅

𝑇𝑇
∆𝑆𝑆 ,
𝐶𝐶t m

(13)

where C H=T(S / T)H and C t are the heat capacity in the86

Isothermal magnetic entropy change,Sm (J/kg K)

0
-5
-10
-15
-20
-25
-30
180

La(Fe0.88Si0.12)13
TC = 195 K

0-1T
0-2
0-3
0-4
0-5

190
200
210
220
Temperature, T (K)230

Fig. 5 Temperature dependence of the isothermal
magnetic entropy change S m for La(Fe0.88Si0.12)13
compound44).
10Adiabatic temperature change,
Tad (K)

La(FexSi1-x)13 compounds (space group Fm3̅𝑐𝑐 ) have aferromagnetic ground state.
In the concentration
0.86≤x≤0.90, the field-induced first-order magnetictransition between the ferromagnetic (FM) andparamagnetic (PM) states, that is, the itinerant-electron
metamagnetic (IEM) transition, takes places above theCurie temperature TC. We have pointed out that this
IEM transition causes a large isothermal magnetic
entropy change S m and a large adiabatic temperaturechange Tad44).
In principle, the magnetic cooling system is equivalent to conventional gas cooling systems. That is,thermodynamic parameters V (volume) corresponds to–M (magnetization) and P (pressure) is changed by H
(magnetic field). Therefore, the Maxwell relation is
given by the following expression:

La(Fe0.88Si0.12)13
TC = 195 K

8

0-1T
0-2
0-3
0-5

6
4
2
0
180

190
200
210
220
Temperature, T (K)230

Fig. 6 Temperature dependence of the adiabatic
temperature change Tad for La(Fe0.88Si0.12)13
compound44).
magnetic field and the total heat capacity, respectively.
Figure 6 shows the temperature dependence of Tad in0H =1–5 T for La(Fe0.88Si0.12)13 compound44). The
value of Tad exhibits a large peak above TC and the
maximum value of Tad is 8.6 K in 0H=5 T at 195 K.
The value of TC of La(FexSi1-x)13 compounds isincreased significantly by hydrogen absorption,accompanied by a marked volume expansion44). The
cubic NaZn13-type structure is kept after hydrogen
absorption. Furthermore, the thermomagnetization
curves still exhibit a significant magnetization changearound TC, because the thermal-induced first-ordertransition is maintained after hydrogen absorption44).
The magnitude of M /T around TC=323 K for thecompound with hydrogen concentration y =1.5 is almost
the same as that of the compound with y =0.0 and
TC=195 K. After annealing at about 400 K, TC is hardly
changed, and hence the desorption of the hydrogen for
La(Fe0.88Si0.12)13Hy compounds scarcely proceeds below
400 K, because TC is significantly sensitive to y.

Journal of the Magnetics Society of Japan Vol.41, No.5, 2017

2.4.2 Isotropic volume magnetostriction
It has been pointed out theoretically that the onset ofthe magnetic moment accompanied by the IEM
transition results in a volume change due to themagnetovolume effect.
For x=0.88, the magnetic
moment caused by the IEM transition exceeds l B andresults in a huge positive volume magnetostriction of
about l.5% just above TC47). The volume magnetostriction s of the IEM transition is due to the onset ofmagnetic moment caused by exchange splitting of
3d-electron bands and given as48)
𝜔𝜔s (𝑇𝑇) = 𝜅𝜅𝐶𝐶mv {𝑀𝑀(𝑇𝑇)2 + 𝜉𝜉p (𝑇𝑇)2 },(14)

where  is the compressibility and Cmv is themagnetovolume coupling constant. The mean squareamplitude of spin fluctuation is given by p(T ) 2 . The
origin of the huge magnetostriction in La(FexSi1-x)13
compounds is correlated with electronic state of
3d-electrons of Fe, different from the single-ion
magnetoelastic coupling in TbFe2-based magnetostrictive materials.
Therefore, it is expected that
La(FexSi1-x)13 compounds are candidates as new-typehigh performance magnetostrictive materials. It isnotable that the transition field of the IEM transition at
TC equals zero and increases with temperature.
The maximum value of thermal expansivity, L / L, of
La(Fe0.88Si0.12)13 compound greatly exceeds the value ofL / L of the polycrystalline TbFe2 compound. As is wellknown, TbFe2-based magnetostrictive materials exhibit
a large ⫽–⊥, while the volume change ⫽+2⊥is nearlyzero and the magnetostrictive properties are anisotropic.
These anisotropic magnetostrictive properties of TbFe 2
compound are mainly originated from the rotation of thelocalized magnetic moment of Tb.  The polarization of
3d-electron bands causes a volume change proportional
to the square of the local magnetization as given Eq. (14)and not only ⫽–⊥ but also ⫽+2⊥ has finite values. A
spontaneous volume change of 1.2 % occurs sharply at TC
and the value of three times of L / L is very close to thatof the volume change at TC. Therefore, these results
indicate that the volume magnetostriction is due to thetransition from the PM phase with a small volume to the
FM phase with a large volume and its volume change isalmost isotropic47).
Such isotropic magnetostrictive
properties are very attractive compared to those of
TbFe2-based anisotropic magnetostrictive materials,
because a huge magnetostriction can be obtained even ina poly-crystalline state.
Accordingly, no crystallographic texture control such as directional solidification
is essential.
It is necessary to adjust the Curie

temperature in order to obtain a huge magnetostriction
in low fields around room temperature. It is notablethat an isotropic giant linear magnetostriction due tothe IEM transition for La(FexSi1-x)13Hy compounds isalso observed in the vicinity of room temperature 49).
2.5 Antiferromagnetism, magnetocrystalline anisotropy
and exchange coupling in Mn alloys
2.5.1 Mn rich disordered (-Mn) and ordered L12 alloys
Practically, the giant magnetoresistance (GMR) andtunnel magnetoresistance (TMR) are very important forelectronic and magnetic devices such as magnetic
recording heads and magnetic random access memory(MRAM) cells. There are many systems having acomplex (non-collinear) spin structure, depending on thed-electron number and the crystal structure associatedwith the geometrical frustration of magnetic moments.
Exchange-biasing characteristics and the blocking
temperature are closely correlated with the spin
structures and the magnitude of the Néel temperature
TN.

1000
Néel temperature, TN (K)Furthermore, tuning TC by adjusting constituent
elements, we can set up magnetic refrigerator systems ina wide working temperature range from 60 to 340 K46).
From these data, it is concluded that La(FexSi1-x)13 andtheir modified compounds are promising magnetic
refrigerants.

Ru

Mn100-xTMx
-phase (DO)

800
Ir
Rh

600 -Mn

200

Ni

Cu

400

0

Fe
Pt

Pd
10
20
30
Concentration, x (at. %)40

Fig. 7 Concentration dependence of the Néel
temperature, TN, for the disordered  (fcc)-phase
Mn100-xTMx alloys (TM=Ir, Ru, Rh, Ni, Fe, Pt, Pd
and Cu)7,8,50).
The  (fcc)-phase disordered (DO) Mn-based alloys arestabilized by several kinds of additional elements inanalogy with Fe-based -phase alloys. The concentration
dependence of TN of Mn100-xTMx -phase disordered alloy
systems (TM: Transition Metal) is gathered in Fig. 77,8,50).
The value of -Mn pure metal is added in the same figure,for comparison. The lower the electron concentration,
and also the larger the atomic size of additive elements
such as Ru, Rh and Ir, the higher the Néel temperature
TN becomes. What has to be noticed is that themagnitude of for the disordered -phase Mn–Rh and
Mn–Ir alloys is lower by 150‒200 K than that for ordered
L12-type (≡ Cu3Au-type) counterparts, depending on thealloying element and its composition.

Journal of the Magnetics Society of Japan Vol.41, No.5, 201787

In the 1Q and 2Q SDW structures in  (fcc)-phase
disordered Mn-based alloys, the magnetic moment points
to the parallel and the perpendicular to the c-axis,
respectively. Take heed that the angles  and  aredefined individually for each site. When  is settled at
45 degrees,  =0,  = 90 and  = cos-1(1⁄√3)= 54.7 degrees
correspond to the so-called 1Q, 2Q and 3Q SDW
structures (see the top panel of Fig. 9), respectively,
which are labeled multiple-Q SDW structures. In the-phase disordered Mn alloys, the 1Q, 2Q and 3Q SDW
structures are observed by changing temperature and/or
composition7,8,50).
2.5.2 L10-type Mn alloys and exchange coupling
In the vicinity of the equiatomic concentration, Mn
forms alloys with Ni, Pd, Pt, Rh and Ir in a wide range ofconcentration. The crystal structure of these alloys is a
B2 (CsCl)-type cubic phase at high temperatures andtransforms into an L10 (CuAu-1)-type tetragonal phasewith a diffusionless martensitic transformation process
at low temperatures8,50). The L10-type phase has acollinear antiferromagnetic structure (AF-1) and MnNi,
MnPd and MnPt equiatomic alloys indicate a very highNéel temperature TN of about 1100, 780 and 970 K,
respectively8,50).
Due to such a high stability of
antiferromagnetism, especially, MnPt alloy has been
investigated intensively from the viewpoint of practicalapplications as antiferromagnetic pinning layers of GMR
and TMR devices. Moreover, the linearized muffin-tin
orbitals (LMTO) band calculations including thespin-orbit interaction were performed for MnPt in order
to investigate the magnetocrystalline anisotropy energy(MAE) because it plays an important role in theexchange-bias field in spin valves. By using the force
theorem, the MAE is obtained from the following
expression:
E MAE=E [100]–E [001],(15)

where E [n] is the sum of the eigenvalues of theKohn–Sham equation52) when the magnetic moment isparallel to the crystal axis [n] in the L10-type structure.
The magnetocrystalline anisotropy constant of MnPt
equiatomic alloy is calculated to be about 1.39 x106 Jm-3,
larger in magnitude with the positive sign larger thanthat with the negative sign of MnNi equiatomic alloy51).
Plotted in Fig, 8 is the concentration dependence of TN of
L10-type ordered MnIr, MnNi, MnPt and MnPd alloys8,50).
The MnIr alloy system exhibits the highest value of TN in
the whole concentration range. In addition, attention
should be paid to that the concentration dependence of
TN for this alloy system is not so sensitive, comparedwith that of the others of L12-type ordered alloy systems.
The relationship between the spin structure and themagnetization loop is provided in Fig, 953). The top
panel represents the magnetic primitive cells constitutedof four atoms in the  (fcc)-phase structure. The
presence of non-magnetic atoms changes the number ofinteracting spins at the nearest neighboring atomic sites,
causing the spin frustration in the -phase disordered
alloy system. This spin frustration lowers the energy ofthe 3Q spin structure, compared with that of thecollinear spin structure54). The initial spin configuration is assumed to be in the 1Q structure, and hence
the final solution is obtained to be the 3Q spin structurein the case of the disordered alloy with 75% magnetic
atoms and 25% non-magnetic atoms.
The spin
configuration of the ordered AFM layer having theL10-type lattice structure was also calculated by using
the Monte Carlo method. Two magnetic and two
non-magnetic atoms are included in the magnetic unit
cell. The spin correlation angle between the nearest
neighboring spins indicates 180 degrees and it
corresponds to the AF-1 spin structure, being a typicalcollinear spin structure of AFM alloys.
The magnetization process in an external field was
calculated by adding the FM layer to such AFM layers.
The motion equation of spins was solved directly within
the framework of the classical Heisenberg model53).
ℋ = − ∑(i,j) 𝐽𝐽1ij (𝑺𝑺i ∙ 𝑺𝑺j ) − ∑(i,k) 𝐽𝐽2ik (𝑺𝑺i ∙ 𝑺𝑺k )− ∑ 𝐷𝐷i (𝑺𝑺i ∙ 𝒏𝒏i )2 − g𝜇𝜇B ∑(𝑺𝑺i ∙ 𝑯𝑯app ),iFig. 8 Concentration dependence of the Néel
temperature, TN, for the L10-type Mn100-xTMx alloys(TM: Ir, Ni, Pt and Pd))7,8,50).

88

i

(16)

where the unit vector S i denotes the spin at the i th atom,and the summation is made over all possible spin pairs
by using the exchange constants, J1 and J2. The indicesj and k respectively represent the first and second
nearest neighbo ring s pi ns for i th a tom. The
summations for j and k are firstly carried out with fixed i,and later on, the summation for i is carried out on all
atoms. The spin pairs i, j> and i, k> represent thesummations by such the procedure. The third andfourth terms in Eq. (1) describe the magnetic anisotropy
energy and the Zeeman energy, respectively. The
g -factor is given by g , D i , and the unit vector n i ,respectively stand for the anisotropy applied field isJournal of the Magnetics Society of Japan Vol.41, No.5, 2017

given by H app . The magnetic easy axis n and theapplied field H app point to the [011] and the [21̅1̅]
directions, respectively. As seen from the figure, theAFM/FM bilayer with the 1Q, 2Q or AF-1 structures only
displays the coercivity in the magnetization loop without
any shift. That is, only the 3Q structure can realize theloop shift caused by the unidirectional exchange-bias
field 53) . At first glance, the above-mentioned modelseems to be invalid for the formation of unidirectional
exchange-bias field in the collinear systems. As awhere P 1 and P 2 are the spin polarization ratio ofelectrodes55). The spin polarization ratio P is obtained
from the following expression56):
𝑃𝑃(%) = |𝑁𝑁↑ (𝐸𝐸F ) − 𝑁𝑁↓ (𝐸𝐸F )| x100,𝑁𝑁↑ (𝐸𝐸F ) + 𝑁𝑁↓ (𝐸𝐸F )(18)

where N↑(EF) and N↓(EF) denote the density of states(DOS) at the Fermi level EF in the majority and theminority spin states, respectively.
In the band
structure of half-metallic compounds the minority bandis semiconducting with a gap at EF, leading to 100% spinpolarization at EF.

L21
2Q

1Q

3Q

(gBHapp/D)

Fig. 9 Spin structures in the magnetic primitive
cell of the  (fcc)-phase lattice and magnetization
loops of the AFM/FM bilayer for 1Q, 2Q, 3Q and
AF-1 spin structures53).
solution to explain existing experimental data, we needto introduce the frustrated spins due to the interfacialroughness and defects into the ordered L10-type AFM
layer. In consequence, the calculated magnetization
loop is actually shifted by the unidirectional
exchange-bias field even in the ordered L10-type alloys.
2.6 Half-metalicity and field induced shape memory
effect in Heusler alloys
2.6.1 Half metal Heusler alloys
The magnetic tunnel junctions (MTJs) are required
for spintronic devices such as magnetic random accessmemories and magnetic sensors.
Half-metallic
ferromagnets (HMFs) with a high spin polarization ratio
P have intensively been investigated as spintronic
devices because MTJs using the HMFs are expected to
exhibit a large value of TMR as given by the following
expression:
𝑇𝑇𝑇𝑇𝑇𝑇 =2𝑃𝑃1 𝑃𝑃2,
1 − 𝑃𝑃1 𝑃𝑃2(17)

Fig. 10 Density of states of the Co2CrGa alloy with
the L21-type structure. The upper and lower
curves in each panel correspond to the majority
and the minority spin states, respectively57).
The DOS calculated by the LMTO method in theatomic sphere approximation (ASA) for the L21 andB2-type (= high temperature phase of L21) structures of
Co2CrGa alloy are sketched in Figs. 10 and 11,
respectively57).
For the calculations of B2-type
structure, the coherent potential approximation (CPA)
method was cooperated. The upper and lower curves ineach figure refer to the majority and minority spin states,
respectively. The DOS of the L21-type structure in Fig.
10 exhibits a half-metal-type electronic structure. As
shown in Fig. 11, the DOS of B2-type is almost
half-metal type and exhibits a high spin polarization.
By using J0 regarded as the effective exchange constant,
𝑇𝑇Ccal is calculated from the following equation within themean-field approximation for spin systems57).
𝑇𝑇Ccal =

2𝐽𝐽0
,
3𝑘𝑘B

(19)

where kB is the Boltzmann constant.
The calculated magnetic moment of each atom M calcal(B/atom), the calculated total magnetic moment 𝑀𝑀tot(B/f.u.), the calculated spin polarization P (%), theJournal of the Magnetics Society of Japan Vol.41, No.5, 201789

alloys with the L21 and B2-type structures are listed inTable 2 57). The saturation magnetic moment M s at 4.2
K is 3.01B / f.u., being consistent to the theoretical value
and the generalized Slater-Pauling line given by

B2

M t =Z t −24,

Fig. 11 Density of states of the Co2CrGa alloy with
the B2-type structure. The upper and lower
curves in each panel correspond to the majority
and the minority spin states, respectively57).
saturation magnetic moment at 4.2 K M s (B/f.u.), theexpexperimental Curie temperature 𝑇𝑇C (K), and thecalcalculated Curie temperature 𝑇𝑇C (K) obtained in themolecular field approximation scheme for Co 2CrGa(20)

where M t and Z t are, respectively, the total values ofmagnetic moment and valence electron numbers in theunit cell. Furthermore, it is clear from the theoretical
calculations that the band structure of B2-type structure
also exhibits a half-metal type and the calculated Curie
temperature TC is about 100 K lower than that of theL21-type structure. Metallurgically, it is noteworthy tonote that the L21-type phase of Co2CrGa is much morestable, compared with that of Co2Cr1-xFexAl58).
In
Co2Cr1-xFexAl alloy system, the systematic theoretical
calculations have been carried out, and pointed out thatthe disordering between the Al and the (Cr, Fe) sites
scarcely degrades the spin polarization58). Conversely,
the disordering between the Co and the (Cr, Fe) sites
brings about a significant reduction of the spinpolarization58).
In fact, a spinodal decomposition
inevitably takes places in Co2CrAl alloy, resulting in
reductions of the saturation magnetic moment and theCurie temperature59).

Table 2 The calculated magnetic moment of each atom M c a l (B/atom), the total magnetic momentcal(B/f.u.), the spin polarization P (%), the saturation magnetic moment M s (B/f.u.) at 4.2 K, calculated
𝑀𝑀tot
exp
and experimental Curie temperatures 𝑇𝑇Ccal and 𝑇𝑇C (K) of Co2CrGa alloys for the L21 and B2-type
57)
structures .

(B/atom)

cal
𝑀𝑀Cr

(B/atom)

cal
𝑀𝑀Gd

(B/atom)

cal
𝑀𝑀tot

(B/f.u.)𝑀𝑀s(B/f.u.)

L21

0.90

1.28

- 0.07

3.01

B2

0.82

1.44

- 0.063.03

2.6.2 Ferromagnetic shape memory alloys
Field-induced strains in ferromagnetic shapememory alloys (FSMAs) attract a great deal of attention
as smart materials. Because FSMAs exhibit both areversible
martensitic
transformation
andferromagnetism, one can control the shape memory
effect by applying magnetic field in addition toconventional controls by temperature and stress.
Field-induced strain in ferromagnetic shape memory
alloys is due to twin-boundary motion and the strain is
tied to the crystallography, not the direction ofmagnetization. That is, it is possible to rotate thedirection of magnetization with no FSMA strain, only
conventional magnetostriction, in FSMAs of relatively
weak anisotropy. In magnetostrictive materials, on theother hand, field-induced strain is a result ofmagnetization rotation relative to the crystallography;

90

exp

cal
𝑀𝑀Co

Structure

P

exp

(%)

𝑇𝑇Ccal
(K)

𝑇𝑇C

3.01

95

419

495

–

84

295

–

(K)

the strain is tied to the direction of magnetization, notthe lattice.
In the martensite phase of FSMAs, the giant
magnetic-field-induced strains (MFIS) have been
observed in Ni2MnGa Heusler alloy60).
Thisphenomenon is concerned with the rearrangement of
twin variants. The MFIS depend on the crystallographic
orientation, because twin-boundary motions are closely
related with both the magnetocrystalline anisotropy and
the Zeeman energies60,61). The martensitic transformation takes place from a L21 (cubic) parent phase toa D022 (tetragonal) martensite phase.
When themartensitic transformation occurs, some variants are
introduced in consequence of the minimum elastic
energy. Furthermore, microscopic magnetic domains
exist in martensite twin variants so as to reduce themagnetic dipole energy. The magnetic domains not

Journal of the Magnetics Society of Japan Vol.41, No.5, 2017

parallel to the direction of applied magnetic field H are
diminished on applying H, and then magnetization Mchanges direction to the magnetic field direction. The
twin variants rearrange and twin boundary is moved inconnection
withtherotational
magnetization,accompanied by macroscopic strains, when themagnetocrystalline anisotropy energy is larger than thedriving energy to rearrange the twin variants.
In order to satisfy the condition for the variant
rearrangement by applying magnetic field, the magnetic
shear stress τ mag should be larger than the mechanicalshear stress τ req required for the variant rearrangement.
Stated differently, the following criterion should besatisfied|𝐾𝐾u |= 𝜏𝜏mag > 𝜏𝜏req ,
𝑠𝑠

(21)

where Ku is the magnetic anisotropy constant62). The
value of τ mag is expressed as |K u|/s, s is the corresponding twinning shear calculated by using the lattice
parameter ratio c/ a of the L10-type martensite phase.
The twinning plane is {101}P for Co–Ni–Ga system,
therefore the twinning shear s is expressed ass={1 −(𝑐𝑐 ⁄𝑎𝑎)2 }.
(𝑐𝑐 ⁄𝑎𝑎 ) (22)

When the magnitude of shear stress becomes τ mag<τ req,
the condition given by Eq. (21) is not satisfied. As aresult, one can obtain no MFIS. In other words, a hightwinning stress is required to induce the twinningtransformation in the L10 structure. Therefore, weneed a static applied twinning stress  tw to assist τ mag.
We have developed a new group of FSMAs in the
Co–Ni–Al B2-based alloy system63,64), which ischaracterized by good ductility and a wide range oftransition temperatures.
Because the composition
range of these FSMAs is located near the B2+ two-phaseregion, these alloys are amenable to the introduction ofthe -phase by proper choice of composition and heat
treatment temperature. The hot-workability and room
temperature ductility of these alloys are significantly
improved by the introduction of the -phase, which is agood advantage for practical applications62).
The
structure of parent phases, the lattice constant ratio c /aof martensite phases and the value of strain of the FSMs
mentioned above are summarized in Table 3.

Table 3 The crystalline structure, the lattice constant ratio c / a, the value of strain of several kinds of
L21- and B2-type ferromagnetic shape memory alloys62,65- 67).
Alloy

Structurec /aStrain (%)

Ni49Fe18Ga27Co6

L21

1.20

-8.5

SA

65

Ni49.5Fe14.5Mn4.0Ga26.0Co6.0

L21

-11.3

SA

66

Co47.5Ni22.5Ga30.0

B2

1.18

-7.6

SA

67

Co41Ni32Al27

B2+

1.16

-3.3

SA, ductile,

62

RemarkReference

SA: stress assisted
3. Closing remarks
Important recent progress has been made in thetheoretical and experimental fields given in Fig.1. It
has been pointed out that the nesting of Fermi surfaces
for electron-hole pairs in Cr is associated with thesuperconductivity for electron-electron pairs, andmathematically the nesting model is identical to the
BCS model for superconductivity. The coexistence of
SDW and superconductivity in various compoundsystems have been reported. Therefore, the discussion
on the relation between SDW and superconductivity
would become active68). The concentration dependenceof Cr-based alloys is in diverse ways, depending on
additives. The formation of local moments such as Fe,
Co, or Ni, might strongly modify the behavior of Cr anddecreases TN9). However, strange that Cr-Fe-Co ternary
alloys increase TN as discussed in connection with Fig. 3.
Since the discovery of giant magnetoresistance inmultilayers of Fe/Cr, great effort has been done inunderstanding the magnetic coupling through chromium
antiferromagnetic layers69).
Many data are very
intricate because the SDW state depends on manyfactors such as temperature, film thickness, interface
exchange coupling, hybridization with the adjacentlayers and interface roughness.
A finite-temperature theory of magnetism that takes
into account the fluctuations of local magnetic moments
due to structural disorder has been discussed on thebasis of the functional-integral method with the methodof the distribution function70). The results of numericalcalculations point that amorphous iron forms anitinerant-electron spin-glass at low temperatures
because of the nonlinear magnetic coupling between Fe
local moments and the local environment effect on theamplitude of the Fe local moment due to the structural
disorder. It should be emphasized that the calculated
spin-glass friezing temperature (120 K) is in good

Journal of the Magnetics Society of Japan Vol.41, No.5, 201791

agreement with our data on Fe-rich amorphousalloys3,27).
Magnetic properties of three dimensional Al-based
icosahedral quasicrystals are presented in Section 2.3.
Two dimensional decagonal, octagonal and dodecagonal
alloys have also been known as quasicrystals. However,up to the present date, no practical magnetic
quasicrystals have been reported, although some twodimensional quasicrystals exhibit excellent photonic
properties71). Since quasicrystals are in both covalentand metallic states without lattice periodicity,
fundamental properties of these quasicrystals haveattracted much attention in the fields of strongly
correlated electron and frustration systems.
Imry and Wortis discussed the broadening of thefirst-order transition due to local fluctuations caused byan intrinsic frozen disorder45).
The extent ofbroadening is affected by the size of frozen correlated
region measured by the coherence length, accompanied
by a spatial distribution of the phase transition field andtemperature. Such frozen disorders are easily caused
in multi-component materials. Therefore, the Maxwell
relation is valid to calculate S m for metamagnetic
La(FexSi1-x)13. In fact, its magnitude obtained from theMaxwell relation is comparable to that from theClausiusClapeyron equation. This model would also
explain the Invar characteristics given Figs. 2 and 3 as acomplete rounding of the first-order transition of Cr-Fe
and Cr-Si alloys by addition of the third element.
Antiferromagnetic spintronics has become an active
field of research. Park et al. have demonstrated a large
spin-valve-like signal in a NiFe/Mn80Ir20/MgO/Pt stackwith an antiferromagnet on one side and a nonmagnetic
metal on the other side of the tunnel barrier.
Ferromagnetic moments in NiFe are reversed byexternal fields of approximately 50 mT or less, and theexchange-spring effect of NiFe on Mn80Ir20 -phase alloy(see Fig. 7) induces rotation of antiferromagnetic
moments, which is detected by the measured tunnelinganisotropic
magnetoresistance.
Their
work
demonstrates a spintronic element whose transportcharacteristics are governed by an antiferromagnet72).
There are four kinds of Heusler-type alloys, i.e., XYZ(C1b), X2YZ (L21), inverse Heusler and quaternary
Heusler and a number of half metal Heusler alloys havebeen reported.
The structural stability comes toattention from practical viewpoints. The sp-elements
are not responsible for the existence of the minority gap;
whereas, they are nevertheless very important for thephysical properties of Heusler alloys because
hybridizations of d–ps electrons contribute to thebonding state73). Therefore, studies on the chemical
orbital Hamiltonian population are considered to beimportant.
Acknowledgements Accomplishing my research described in this article would not have been possible
without many supports. I am indebted to all our past

92

collaborators, postdocs, and students, too numerous tolist individually. Their contributions are clear from thelarge number of citations of their work in many
bibliographies. I acknowledge the generous financial
supports from the Ministry of Education, Culture, Sports,
Science, and Technology of Japan and from the Japan
Science and Technology Agency.
References
1) K. Fukamichi: Magnetic Properties of Metals III, 19 I·1 (ed.
H. P. J. Wijn, Landolt-Börnstein Springer-Verlag, Berlin,1994), pp. 193-213.
2) K. Fukamichi: Amorphous Metallic Alloys (ed by F. E.
Luborsky, Butterworths Co. Ltd., London, 1983), pp.
317-340.
3)
K.
Fukamichi:
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Received March 5, 2017; Accepted May 15, 2017

Journal of the Magnetics Society of Japan Vol.41, No.5, 201793