Buschow K.H.J. (Ed.) Concise Encyclopedia of Magnetic and Superconducting Materials

Подождите немного. Документ загружается.

down-spin bands, can be neglected under ordinary

conditions (at least at low ﬁelds). Note that the spin

component could be basically viewed as arising from

the two-carrier (up- and down-spin carriers) model

with different relaxation times described above. The

skew scattering mechanism can be understood within

the classical picture as a deﬂection of the electron

trajectory by an angle y after the scattering through

spin–orbit coupling. The side jump mechanism is ba-

sically a quantum effect, which can be viewed as a

side step of the center of gravity of a wave packet, as

shown in Fig. 4, as an analogy.

The difference in the two mechanisms is reﬂected in

the dependence of each term on the resistivity r

(Hurd 1972): the anomalous Hall coefﬁcient R

is the

sum of the skew scattering term proportional to r

and the side-jump term proportional to r

¼ ar þ br

ð12Þ

The electron scattering on only static imperfections

(like impurities) was initially considered as the origin

of the anomalous Hall effect in measurements of the

impurity concentration dependence of R

, where the

residual resistivity r

is employed in Eqn. (12).

Later on, it has been extended to the scattering by

thermal spin disorder and/or by phonons, both ex-

perimentally and theoretically (Irkhin and Shavrov

1962, Kondorskii 1965). In most theories, the mag-

netic part of the resistivity r

mag

is assumed to be

inserted in Eqn. (12), although it is not always easy to

separate the nonmagnetic contribution from the total

resistivity. In rare-earth intermetallic compounds,

mag

is usually obtained by subtracting the resistivity

ref

(T) of nonmagnetic reference compound from the

total resistivity r

tot

(T) as:

mag

ðTÞ¼r

tot

ðTÞr

ref

ðTÞð13Þ

It should be noted, however, that for the itinerant

magnetic electrons model, the relation, R

tot

, was

theoretically reported for the thermal phonon or spin

disorder scattering at elevated temperatures (Hurd

1972).

(b) Kondo systems

The skew scattering theory based on the Coqblin–

Schrieffer model (1969) was ﬁrst proposed by Fert

(1973), and was successfully applied to explain the

Hall effect in dilute lanthanum–cerium alloys. For

cerium compounds, Coleman et al. (1985) have given

an explanation of the large anomalous Hall constant

in terms of skew scattering by impurities, based on

the resonance-level model assuming a strong spin–

orbit coupling in the two limits of TbT

and T5T

Later, Fert and Levy (1987) formulated the more

generalized form, which describes experiments rea-

sonably well in a wider temperature range from

T4T

to T5T

. The Hall coefﬁcient from the in-

trinsic skew scattering was given by the following

general expression:

ðTÞ¼gw

red

ðTÞr

ðTÞð14Þ

where w

red

(T){¼3k

w/(gm

)

j(j þ1)} is the reduced

susceptibility, r

(T) is the resistivity due to the res-

onant scattering. The coefﬁcient g takes different val-

ues g

and g

above and below T

, related to the

phase shifts d

for channel l:

¼ð5=7Þgm

1

sind

cosd

ð15aÞ

for TbT

, and

¼ð5p=21Þgm

1

sinð2d

 d

Þsind

=sin

ð15bÞ

for TbT

Fert and Levy (1987) have succeeded in reproduc-

ing the experimental temperature dependence of R

in typical heavy-Fermion compounds such as CeAl

and CeCu

. Even for the coherent state, they found

that Eqn. (14) accounts for the main features of the

Hall effect.

For the coherent state, Kontani and Yamada

(1994) ﬁrst derived a formula for the anomalous Hall

coefﬁcient, based on the Fermi liquid theory. The

anomalous part of the Hall coefﬁcient is described as:

¼ Cw

enh

ð16Þ

Figure 4

Schematic comparison of skew-scattering and side-jump

scattering.

Figure 3

A typical example of Hall resistivity in a ferromagnetic

thin plate.

330

Intermetallics: Hall Effect

where the electrical resistivity follows r ¼r

þAT

and w

enh

is the enhancement factor proportional to

the magnetic susceptibility (see Electron Systems:

Strong Correlations).

3. Typical Experimental Results

Firstly, the most important factors for analyzing the

temperature dependence of Hall coefﬁcient in Kondo

systems can be summarized as follows.

From the viewpoint of reliability of the analysis,

(T ), w(T ), and r(T ) should be measured on the

same sample and with the same geometry, which

reduces the errors due to unknown sources. For

compounds with an anisotropic crystal structure,

measurements on single crystals are essential. The

magnetic contribution to the measured resistivity

exp

(T ) could be estimated by subtracting the resis-

tivity r

ref

(T ) of a proper reference compound (for

example, a lanthanum analog for a cerium com-

pound) as:

mag

ðTÞ¼r

exp

ðTÞr

ref

ðTÞð17Þ

Using r

mag

(T ) thus obtained, R

(T ) is plotted

against w(T ) r

mag

(T ) as shown in Fig. 5. The slopes

in the two linear regions above T

and below T

give

and g

in Eqns. (15a) and (15b).

3.1 Well-localized 4f Compounds

The series RECu

(RE: La, Pr, Nd, and Sm) was

selected as a typical example for the well-localized

system. Fig. 6 shows the temperature dependence of

in RECu

for the light rare-earth elements (



Onuki

et al . 1989). At high temperatures, R

is only weakly

temperature-dependent for all the compounds except

CeCu

. For all the compounds, R

is approaching

5 10

10

1

, which should be the normal part

of the Hall coefﬁcient. The negative sign of R

sug-

gests that the relaxation time for the carriers with

electron-like Fermi surface curvatures, positive r

1

Eqn. (7), is larger than that for the hole-like Fermi

surface curvatures, since RECu

is a compensated

metal with the same volume for both electron-like

and hole-like Fermi surfaces.

for NdCu

and SmCu

shows a clear change at

the corresponding Ne

el temperatures T

of 6 K and

9 K, respectively. At T

, two possible contributions

coexist; namely the change in the carrier concentra-

tion resulting from the AF super-zone gap formation,

and the change in anisotropy in the relaxation time.

For SmCu

, taking into account the fact that the

change in R

is steep, while that in r is gradual, the

two contributions compete in r and cooperate in R

The unique behavior in CeCu

results from the

dominant Kondo scattering in this compound, which

will be discussed below.

3.2 Cerium–Kondo Systems at Higher Temperatures

(Ce

1x

, CeRu

, etc.)

Among the rare-earth intermetallic compounds, the

works on the Hall effect were rather limited to the

Kondo or heavy-Fermion compounds, mainly cerium

and ytterbium compounds (Cattaneo 1985,



Onuki

et al. 1993). Here, we concentrate on the current un-

derstanding of the Hall effect in cerium compounds,

based on typical examples in correlation with the

reported theories.

To start with, the Hall coefﬁcient of Ce

1x

which has been the most systematically investigated,

is best suited for comparison with the skew scatter-

ing theory. Figure 7 shows the comparison of the

temperature dependences of R

in the experiment

with Eqn. (14) for selected values of cerium con-

centration. Here, the magnetic contribution to the

resistivity, r

(T ), has been estimated by subtracting

the resistivity r

(T ) of a LaCu

sample from the

Figure 6

Temperature dependence of R

in RECu

Figure 5

Schematic ﬁgure showing how to estimate g

and g

from

the experimental R

331

Intermetallics: Hall Ef fect

experimental one: r

exp

(T ); r

(T ) ¼r

exp

(T )r

(T ).

red

¼w/C is estimated by using the experimental val-

ue of w. It should be noted that the experimental

curves could be ﬁtted by almost the same value of the

phase shift d

E0.02 for all cerium concentrations.

The agreement is quite good over a wide temperature

range. Even for pure CeCu

, the agreement is satis-

factory above B20 K, below which the development

of the coherent state naturally leads to a different

conduction electron state. However, it is not easy to

separate the intrinsic skew scattering contribution

only from this ﬁgure, since a large anisotropy in R

at low temperatures could be expected.

For the coherent state, the scaling relation

Eqn. (16) has been tested in selected heavy-Fermion

systems. The results for CeCu

and CeRu

are

shown in Fig. 8, where the linearity is reasonably

good for the lowest temperatures (



Onuki et al. 1993).

For CeRu

, there has been a controversy over

the difference in the anisotropy between the anom-

alous Hall coefﬁcient and the magnetic susceptibility:

the anisotropy in R

is small in contrast to the large

anisotropy in w, reﬂecting the crystalline electric ﬁeld

(CEF) of the 4f levels. This fact opens the question of

whether the temperature dependence of R

is caused

by the skew scattering contribution to the anomalous

Hall effect or not. Kontani et al. have explained

the difference in the anisotropy as being due to the

different contribution from the higher excited CEF

levels to w and R

(Kontani et al. 1997). However, in

order to fully understand the heavy-Fermion state,

more systematic investigations on high-quality single

crystals are necessary.

3.3 Uranium Intermetallic Compounds

(UNiGa, UCoAl, UPd

)

UNiGa is one of the typical compounds whose

Hall effect has been successfully utilized to solve

important issues (Kobayashi et al. 1996). At low

temperatures, uranium moments order ferromagnet-

ically within the basal plane sheets with a strong

uniaxial anisotropy along the c-axis. An antiferro-

magnetic (AF) structure, characterized by c-axis

stacking of the type (mmkmkk), is stabilized.

A magnetic ﬁeld applied along the c-axis induces a

ﬁrst-order type metamagnetic transition leading to a

ferromagnetic (F) stacking, accompanied by a large

resistance change across the metamagnetic transition

Figure 7

Comparison of R

in the experiment with Eqn. (14) for La

1x

Figure 8

versus r

plots for CeCu

and CeRu

at low

temperatures.

332

Intermetallics: Hall Effect

(Sechovsky et al. 1991). Spin-dependent scattering

was initially thought to be the origin of the large

magnetoresistance, since it bears a geometrical re-

semblance to the giant magnetoresistance (GMR) in

artiﬁcial magnetic superlattices.

Another plausible explanation of the large magne-

toresistance is based on the Fermi surface reconstruc-

tion due to the disappearance of the AF super-zone

gap. This was initially rejected from results of specific

heat measurements (Sechovsky et al. 1993), where no

clear change in C/T was observed across the meta-

magnetic transition.

The measurement of Hall effect has been used as a

powerful tool to judge the origin of the large mag-

netoresistance (Kobayashi et al. 1996). Figure 9

shows the ﬁeld dependence of R

across the meta-

magnetic transition at 4.2 K, along with the magnet-

ization and magnetoresistance measured on the same

sample in the same geometry. Fortunately, R

practically ﬁeld-independent at low and high ﬁelds,

except close to the metamagnetic transition, since

both M and r are almost constant due to the strong

uniaxial anisotropy. Therefore, the normal Hall

coefﬁcient R

below and above the metamagnetic

transition can be determined from the slopes in the

two regions. R

changes from 2.9 10

9

1

the AF state to 1.7 10

9

1

in the F state,

which gives evidence of a Fermi surface change across

the metamagnetic transition. Based on this fact, the

specific heat was remeasured, and a clear change of

the g value was found from 43 mJ K

2

mol

1

below to

48 mJ K

2

mol

1

above the metamagnetic transition.

This corroborates the scenario of the Fermi surface

reconstruction across the metamagnetic transition.

Another important issue on this material is the

itinerancy of 5f electrons. The temperature depend-

ence of R

for UNiGa is shown in Fig. 10, along with

those of r(T) and M(T).

It is meaningful to compare this result with those

obtained on UPd

. The latter compound was con-

ﬁrmed to be a typical 5f-localized system, based on an

inelastic neutron scattering experiment. The temper-

ature dependences of R

for B:[0001] and [1010]

are shown in Fig. 11 (



Onuki et al. 1993). One promi-

nent feature of these ﬁgures is the large difference in

the temperature-dependent part of R

between the

two compounds. That in UNiGa is about two orders

of magnitude larger than that in UPd

. This fact

strongly suggests that the 5f electrons responsible for

the magnetism are itinerant in UNiGa, while they are

localized in UPd

Figure 9

Field dependence of r

, r, and M in UNiGa at 4.2 K.

Figure 10

Temperature dependence of r, R

and M in UNiGa.

Figure 11

Temperature dependence of R

in UPd

333

Intermetallics: Hall Ef fect

In the paramagnetic state, R

for UNiGa varies

linearly as rM, as expected from Eqn. (14), while in

the ferromagnetically aligned state at low tempera-

tures, R

/r is linear, passing through zero as a func-

tion of r (Fig. 12). Namely, in the high ﬁeld state, the

side jump scattering in Eqn. (12) dominates, resem-

bling 3d ferromagnetic materials with a strong elec-

tron scattering. It should be also noted that such

good linear relations were found experimentally by

using the total resistivity, as was theoretically ob-

tained for the itinerant magnetic system (Irkhin

and Shavrov 1962), in many uranium compounds

(without subtracting the nonmagnetic component).

In contrast, no simple correlation was found be-

tween R

, w, and r for UPd

, as shown in Fig. 13,

which can be understood if the contribution from the

temperature-dependent anisotropic relaxation time

masks a small skew scattering contribution. This fact

is another evidence of the localized character of the 5f

electrons (at least there is only very little hybridiza-

tion) in this compound.

Another example of 5f-itinerant compounds is

UCoAl (Sato et al. 1998), which is a rare example

exhibiting a metamagnetic transition from the para-

magnetic ground state. The mechanism of the meta-

magnetic transition in typical 3d metamagnets, such

as YCo

and LuCo

, is well understood in terms of

itinerant-electron metamagnetism resulting from the

high 3d-band density of states near the Fermi level

(Goto et al. 1994). A close resemblance of UCoAl to

the 3d-itinerant metamagnets has been invoked,

based on intensive studies of the effect of alloying

and pressure on the metamagnetic transition. Figure

14 shows the temperature dependence of R

for two

different values of the magnetic ﬁeld. The inset shows

M/B measured on a same sample. The temperature

dependences of the two properties resemble each

other, which becomes more apparent in the R

vs. wr

plot in Fig. 15. In the wide temperature range above

60 K, R

is linear against wr, as expected for the skew

Figure 12

(a) R

versus rM plot and (b) R

/r versus r plot for

UNiGa.

Figure 13

Temperature dependences of R

, w, wr in UPd

Figure 14

Temperature dependence of R

in UCoAl.

Figure 15

versus wr plot for UCoAl. The inset shows r

versus

M plot at high ﬁelds.

334

Intermetallics: Hall Effect

scattering mechanism. This fact, as well as the large

magnitude of R

, suggests the 5f electrons to be

itinerant.

The ﬁeld dependence at low temperatures in

Fig. 14 reﬂects the metamagnetic transition. Figure

16 shows the ﬁeld dependence of resistance, Hall re-

sistivity and magnetization for UCoAl on the same

sample and under the same conditions. We note an

apparent resemblance between the Hall resistivity

and the magnetization curves. Note that in the ﬁeld-

induced ferromagnetic state, r

varies as r

(the inset

of Fig. 15). Across the metamagnetic transition

ﬁeld, the transverse magnetoresistance shows a clear

decrease with increasing ﬁeld, while an increase in

the resistivity has been reported for the longitudinal

geometry. This can be understood to be a so-called

anisotropic magnetoresistance resulting from the an-

isotropy in the spin–orbit coupling, which is common

in 3d-ferromagnetic metals and alloys. This fact also

suggests that UCoAl is in a 5f-band split ferromag-

netic state above the metamagnetic transition ﬁeld.

The change in the density of states due to

the band splitting was further manifested in the ﬁeld

dependence of specific heat as a decrease of C/T

(Matsuda et al. 1999).

4. Concluding Remarks

The Hall effect in intermetallics is not always analy-

zed straightforwardly. However, depending on the

system, it has the potential to give important infor-

mation on the electronic structure, which can hardly

be extracted by other experimental techniques. In

order to accomplish this, a proper analysis including

the separation of different contributions, e.g., either

normal or anomalous, is essential.

See also:5f Electron Systems: Magnetic Properties;

Heavy-fermion Systems; Intermediate Valence Sys-

tems; Kondo Systems and Heavy-fermions: Trans-

port Phenomena

Bibliography

Berger L, Bergmann G 1980 The Hall effect of ferromagnets.

In: Chien C L, Westgate C R (eds.) The Hall Effect and its

Applications. Plenum, New York, pp. 55–76

Cattaneo E 1985 Hall effect of some intermediate-valent Ce and

Yb compounds. Z. Phys. 47/48, 529–31

Chambers R G 1976 Transport properties: surface and size ef-

fect. In: Ziman J M (ed.) The Physics of Metals. Cambridge

University Press, London, pp. 175

Coleman P, Anderson P W, Ramakrishnan T V 1985 Theory

for the anomalous Hall constant of mixed-valence systems.

Phys. Rev. Lett. 55, 414–7

Coqblin B, Schrieffer J R 1969 Exchange interaction in alloys

with cerium impurities. Phys Rev. 185, 847–53

Fert A 1973 Skew scattering in alloys with cerium impurities.

J. Phys. F: Met. Phys. 3, 2126–42

Fert A, Friederic A, Hamzic A 1981 Hall effect in dilute mag-

netic alloys. J. Magn. Magn. Mater. 24, 231–57

Fert A, Hamzic A 1980 Hall effect from skew scattering by

magnetic impurities. In: Chien C L, Westgate C R (eds.)

The Hall Effect and its Applications. Plenum, New York,

pp. 77–98

Fert A, Levy P M 1987 Theory of the Hall effect in heavy-

fermion compounds. Phys. Rev. B 36, 1907–16

Goto T, Aruga Katori H, Sakakibara T, Mitamura H,

Fukamichi K, Murata K 1994 Itinerant electron metamag-

netism and related phenomena in Co-based intermetallic

compounds. J. Appl. Phys. 76, 6682–7

Hurd C M 1972 The Hall Effect in Metals and Alloys. Plenum,

New York

Hurd C M 1980 ‘‘Pressing electricity’’: a hundred years of

Hall effect in crystalline metals and alloys. In: Chien C L,

Westgate C R (eds.) The Hall Effect and Its Applications.

Plenum, New York, pp. 1–54

Irkhin Yu P, Shavrov V G 1962 Theory of the spontaneous Hall

effect in ferromagnets. Sov. Phys. JETP 15, 854–8

Kobayashi Y, Aoki Y, Sugawara H, Sato H, Sechovsky V,

Havela L, Prokes K, Mihalik M, Menovsky A 1996 Hall

effect and thermoelectric power in UNiGa. Phys. Rev. B 54,

15330–4

Kondo J 1962 Anomalous Hall effect and magnetoresistance of

ferromagnetic metals. Prog. Theor. Phys. 27, 772–90

Kondorskii E I 1965 Role of electron scattering by phonons

and magnetic inhomogeneities in the Hall effect in ferromag-

netic metals. Sov. Phys. JETP 21, 337–41

Kontani H, Miyazawa M, Yamada K 1997 Theory of anom-

alous Hall effect in a heavy-fermion system with a strong

anisotropic crystal ﬁeld. J. Phys. Soc. Jpn. 66, 2252–5

Kontani H, Yamada K 1994 Theory of anomalous Hall effect

in heavy-fermion system. J. Phys. Soc. Jpn. 63, 2627–52

Matsuda T D, Aoki Y, Sugawara H, Sato H, Andreev A V,

Sechovsky V 1999 Specific heat anomaly around metamag-

netic transition in UCoAl. J. Phys. Soc. Jpn. 68, 3922–6



Onuki Y, Yamazaki T, Ukon I, Komatsubara T, Umezawa A,

Kwok W K, Crabtree G W, Hinks D G 1989 Anomalous

Hall coefﬁcient in the f-electron systems—U compounds.

J. Phys. Soc. Jpn. 58, 2119–25



Onuki Y, Yun S W, Satoh K, Sugawara H, Sato H 1993

Anomalous Hall coefﬁcient in the f-electron system. In: Oomi

G, Fujii H, Fujita T (eds.) Transport and Thermal Properties

of f-Electron Systems. Plenum, New York, pp. 103–11

Sato H, Aoki Y, Matsuda T D, Sugawara H, Kobayashi Y,

Andreev A V, Sechovsky´ V, Havela V, Settai R,



Onuki

Y 1998 Transport and thermal properties across phase

Figure 16

Field dependence of r, r

, and M in UCoAl at

4.2 K.

335

Intermetallics: Hall Ef fect

transitions in f-electron systems. In: Kazimierski M (ed.) 11th

Sem. Phase Transitions and Critical Phenomena. Polish Acad-

emy of Sciences, Wroclaw, Poland, pp. 23–33

Sechovsky´ V, Havela L, de Boer F R, Bru

ck E, Suzuki T, Ikeda

S, Nishigori S, Fujita T 1993 Specific heat and magnetoca-

loric effect in UNiGa. Physica B 186–8, 775–7

Sechovsky´ V, Havela L, Jirman L, Ye W, Takabatake T, Fujii

H, Bru

ck E, de Boer R, Nakotte H 1991 Giant magnetore-

sistance effects in UNiGa. J. Appl. Phys. 70, 5794–6

Tsuji M 1958 The thermoelectric, galvanomagnetic and thermo-

magnetic effects of monovalent metals. III. The galvanomag-

netic and thermomagnetic effects for anisotropic media. J.

Phys. Soc. Jpn. 13, 979–86

H. Sato

Tokyo Metropolitan University, Japan



Onuki

Osaka University, Japan

Invar Materials: Phenomena

In the search for a cheaper length standard than the

meter made of Pt–Ir, the Swiss physicist Charles

Guillaume in 1897 discovered ‘‘Invar,’’ a ferromag-

netic (FM) f.c.c. alloy of 65 at.% iron and 35 at.%

nickel with small and almost temperature-independ-

ent—‘‘invariant’’—thermal expansion. After discov-

ering ‘‘Elinvar,’’ a FM f.c.c. Fe–Ni–Cr alloy with

constant elastic behavior, Guillaume was awarded

the Nobel Price for physics in 1920, not the least

because of the technical importance of his ﬁndings.

The understanding of Invar remained a problem

(Wassermann 1990, Wittenauer 1997) before ab initio

calculations (Entel et al. 1993) gave basic insights into

the relations between lattice structure, atomic vol-

ume, and magnetic properties, and brought to light

‘‘anti-Invar’’ and how it relates to martensite, the

long-known structural f.c.c.–b.c.c. transformation of

iron and iron-rich alloys (Wassermann et al. 1999).

1. Basic Physical Properties and Definitions

Invar- and often also Elinvar-like behavior is found

in transition metal alloy systems with different lattice

structures (see Table 1), most with iron as a compo-

nent. Figure 1 shows schematically the most typical

features. The temperature dependence of the thermal

expansion coefﬁcient a (Fig. 1(a)) in the FM ordered

range ToT

(where T

is the Curie temperature) of

Invar is reduced relative to a nonmagnetic reference,

for which a(T) is given by the classical Gru

neisen

theory. Analogous behavior is found in antiferro-

magnetic (AF) Invar alloys such as Fe

in the

range ToT

(where T

is the Ne

el temperature).

Figure 1(b) shows that relative to the Gru

neisen

reference the volume in a FM or an AF Invar alloy in

the range TpT

C,N

is enhanced. This spontaneous

volume magnetostriction, o

, has a maximum value

at zero temperature, o

¼(DV/V )

T ¼0

, correspond-

ing to the area between the data and the reference

curve in Fig. 1(a). Figure 1(c) shows that the shear

modulus, G, starts to soften at T

and then displays

the temperature-independent Elinvar behavior. Fig-

ure 1(d) shows that the bulk modulus, B(T), softens

much above T

in FM Invar alloys, different from

that observed in AF Invar alloys (Wassermann 1990).

For e/ao8.5 (the number of valence electrons per

atom) in f.c.c. iron-based systems, there is also anti-

Invar behavior (Acet et al. 1994), the opposite of

Invar. Anti-Invar behavior describes the increase of

the volume and the enhancement of the thermal

expansion relative to a reference in the high-temper-

ature paramagnetic range. The relative volume in-

crease, DV/V, can be determined quantitatively by

plotting aT

¼f(T/T

), where T

is the melting tem-

perature. For ‘‘normal’’ metals and alloys without

magnetovolume anomalies such a plot results in an

almost universal expansion curve, giving the ave-

rage lattice curve a

lat

(T ), which can be used as

ageneralGru

neisen-like reference for a(T )ofInvar

Table 1

Invar and Elinvar systems.

F.c.c. structure

Fe–Ni

Fe–Pt (ordered)

Fe–Pt (disordered)

Fe–Pd

Fe–Mn

Co–Mn

Fe–Ni–Mn

Fe–Ni–Cr

Fe–Ni–Co

Fe–Ni–Pt, Pd

B.c.c. structure

Cr–Fe

Cr–Mn

Fe–Cr–Mn

Hexagonal structure

Co–Cr

Amorphous

Fe–B, P

Fe–TM (TM ¼Sc, Y, Zr, Hf)

(Fe–TM)Gl (TM ¼Mn, Co, Ni; Gl ¼Si, B, P)

Laves phases, compounds

ETFe

(ET ¼Ti, Zr)

RECo

(RE ¼all except Eu)

(RE ¼Y, Dy, La)

(FeCo)

336

Invar Materials: Phenomena

and anti-Invar alloys. A plot of aT

¼f(T/T

)for

100x

alloys and elemental iron in the f.c.c.

stability range shown in Fig. 2 reveals the continu-

ous transition from FM Invar behavior in Fe

(e/a ¼8.7) to anti-Invar behavior in iron-rich samples

and f.c.c. iron. In Fe

(e/a ¼8.6) both effects are

observed, i.e., Invar behavior for ToT

and anti-

Invar behavior for T4T

.Forx475 (e/ao8.5)

one encounters only anti-Invar behavior, for which an

example is given in the inset of Fig. 2, showing a(T)of

, a paramagnetic alloy with f.c.c. stabil-

ity to 0 k. With decreasing temperature the anti-Invar

behavior terminates at the martensite start tempera-

ture, M

, where the f.c.c.–b.c.c. transition takes place.

Figure 3 shows absolute values of the relative vol-

ume increase, 7DV/V7, vs. the composition (e/a)

caused by the FM and AF Invar effects and the

anti-Invar effect in different f.c.c. alloys. The volume

increases shown occur in addition to the B7.5% in-

crease any metal or alloy experiences in the range

0pTpT

because of the ‘‘normal’’ thermal expan-

sion due to anharmonic lattice potential (Gru

neisen

behavior). FM Invar behavior occurs only for e/a4

8.5, while for e/ao8.5 one ﬁnds either anti-Invar be-

havior at high temperatures and AF Invar behavior

at low temperatures, or anti-Invar behavior at high

temperatures and martensite at low temperatures, if

the f.c.c. stability is limited.

2. Microscopic Understanding

The present understanding of Invar and anti-Invar

behavior stems from ground state properties as de-

termined in ab initio calculations (Entel et al. 1993,

Herper et al. 1999). A simpliﬁed picture is given in

Fig. 4, showing for Invar and anti-Invar the volume

dependence of the total energy E(V) (Figs. 4(a) and

(b)) and the magnetic moments (Figs. 4(c) and (d)).

denotes the equilibrium and V

the critical vol-

ume, where the equilibrium state becomes energeti-

cally unfavorable. In anti-Invar (Fig. 4(a)) the ground

state at V

is a low-spin (LS) state with a small mag-

netic moment (Fig. 4(c)) relative to the FM high-spin

(HS) state at larger volumes, causing an enhanced

anharmonicity of the lattice towards larger volumes

V4V

in E(V). Invar (Fig. 4(b)) is just the opposite,

because the ground state at V

is a FM HS state and

the enhanced anharmonicity in E(V) is towards

smaller volumes VoV

, caused by the LS state.

Figure 2

Thermal expansion coefﬁcient a multiplied by the

melting temperature T

vs. T/T

for f.c.c. Fe–Ni alloys

and f.c.c. g-iron. The upward pointing arrows indicate

the Curie temperatures T

. The curves for 70pxp100

are measured down to their corresponding martensite

start temperatures M

. The solid curve is the

nonmagnetic Gru

neisen-like lattice reference. Inset:

(T/T

) for Fe

anti-Invar, f.c.c. stable to

0K.

Figure 1

Schematic representation of the temperature dependence

of (a) the thermal expansion coefﬁcient a, (b) the relative

volume change DV/V, (c) the shear modulus G, and

(d) the bulk modulus B for an Invar alloy. The dashed

curves are the nonmagnetic Gru

neisen reference.

337

Invar Materials: Phenomena

The presence of these moment–volume instabilities

in Fe–Ni and Fe–Pt Invar alloys has been demon-

strated experimentally in Mo

ssbauer experiments

at 4.2 K, showing HS to LS transitions to occur

when the lattice constant is reduced by high pressure

(Abd-Elmeguid and Micklitz 1989). The ab initio cal-

culations also corroborate the systematic behavior

shown in Fig. 3. With increasing e/a anti-Invar

behavior progressively transforms to FM Invar be-

havior as DE, the energetic difference between the HS

and the LS states, decreases, goes through zero (so

that LS and HS states are degenerate) near e/aB8.3,

and then increases again with the HS state becoming

lower in energy.

Concerning the behavior at ﬁnite temperatures, the

key point is that DE is of the order of milli-Rydbergs

(1 mRyB150 K), so that transitions from the ground

state to the energetically elevated state are possible by

raising the temperature. Thus, as indicated with the

arrows in Fig. 4, in anti-Invar LS–HS and in Invar

HS–LS transitions occur, and the magnetic moment

increases (Fig. 4(c)) or decreases (Fig. 4(d)), respec-

tively. This is conﬁrmed by high-temperature neutron

investigations on the paramagnetic spin ﬂuctuations

of Fe

100x

alloys within the f.c.c. stability range

(Acet et al. 1997). Figure 5 shows the results in a plot

of the average magnetic moment, m, vs. the temper-

ature for compositions between x ¼65 Invar (dashed

curve) and pure g-iron. With increasing iron concen-

tration the slope of the m(T) curves changes sign, in

accordance with a change from a HS–LS transition

for Fe

Invar to a LS–HS transition in Fe

anti-Invar or Fe

, an anti-Invar material

with f.c.c. stability to 0 K. The high-temperature spin

ﬂuctuations in Fe–Ni alloys and in f.c.c. iron are al-

ways FM, although at low temperatures AF corre-

lations occur, even AF long-range order in f.c.c. iron

for T

o70 K. There is evidence from ﬁrst principles

calculations (van Schilfgaarde et al. 1999) that non-

collinear magnetic moments and AF short-range or-

der cause additional lattice softening and negative

anharmonicity in Fe–Ni Invar.

Deeper insight into the microscopic background is

gained from calculations resolving the properties of

the d electrons in the vicinity of the Fermi energy E

(Entel et al. 1993) with respect to their orbital sym-

metry (t

, e

) and their antibonding (AB), nonbond-

ing (NB), or bonding (B) character. The respective

electronic density of states (DOS) curves in the LS

and HS state are shown schematically in Fig. 6. The

Figure 3

Relative volume enhancement 7DV/V7 through FM

Invar, AF Invar, and anti-Invar effect vs. composition in

electrons per atom (e/a) for different f.c.c. alloy systems.

Figure 4

Schematic representation of volume dependence of (a, b)

the total energy and (c, d) the magnetic moment of Invar

and anti-Invar. V

, equilibrium volume; V

, critical

volume for the transition from the LS to the HS state.

Arrows indicate the direction of response of the system

to increasing temperature.

338

Invar Materials: Phenomena

orbitals have a maximum charge density in the

[110] direction and thus in an f.c.c. lattice form strong

dds bonds with nearest neighbors. The t

DOS is

therefore split and shows an energetically high-lying

peak with AB character and an energetically low-

lying one with B character. The e

orbitals have max-

imum charge densities in the [100] direction and in an

f.c.c. lattice form ddp bonds with the next nearest

neighbors. They are NB and found between the two

peaks in the f.c.c. DOS. The position of the Fermi

energy, E

, relative to these levels, which depends on

the electron concentration, the atomic volume, and

the temperature, determines which type of transition

occurs between these levels, to lead to Invar, anti-

Invar, or to a martensitic transformation.

The arrow in Fig. 6 indicates the possible excita-

tions with increasing temperature. For anti-Invar

with a LS ground state the arrow reads from left to

right, because there is a charge transfer from NB mi-

nority-spin e

orbitals to AB majority-spin t

orbit-

als. These excitations lead to a band splitting and

ﬁnally the HS state with a large volume caused by the

AB character of the t

majority-spin states. In Invar

the arrow in Fig. 6 reads from right to left, because

the HS state is the ground state and the charge trans-

fer is from AB t

majority-spin states to NB e

mi-

nority-spin states when the temperature is increased.

This reduces the band splitting, corresponding to a

HS to LS transition.

In summary, more than 100 years after the ﬁnd-

ing by Guillaume, through to the t

–e

scenario, the

microscopic background of Invar, Elinvar, and anti-

Invar and also the relationship to the martensite in

iron-based systems is basically understood. However,

the true solution of the Invar problem will be found

when a ﬁrst principles calculation reproduces cor-

rectly the archetypical property of Invar, i.e., the

anomalous temperature dependence of the thermal

expansion coefﬁcient. To reach this goal a substantial

amount of research is still necessary.