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

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depicted by a periodic arrangement of steps. These

steps reﬂect the band mismatch between the magnetic

and nonmagnetic layers or, in simpliﬁed language, the

different levels of the conduction band in different

metals. The steps have different heights for the two

spin directions, which expresses the exchange splitting

of the majority and minority spins in a ferromagnetic

metal. In Fig. 9 one also sees spin-dependent spikes

which represent random scattering potentials gener-

ated by impurities, defects, or interface roughness. The

spin dependence of the scattering by impurities or de-

fects is well known in bulk magnetic materials (Camp-

bell and Fert 1982) and it is also obvious that the

modulation of the spin-dependent interface steps by

roughness produces spin-dependent scattering poten-

tials. The distribution of both the intrinsic and scat-

tering (or extrinsic) potentials is different in the parallel

and antiparallel magnetic conﬁgurations of the multi-

layers, as illustrated in Fig. 9 by the alternation of

high and low steps or spikes only in the antiparallel

conﬁguration. This dependence of the intrinsic and

extrinsic potential distribution on the magnetic con-

ﬁguration gives rise to the GMR effect and forms the

basis of a number of theoretical models.

5.1 Models of CIP GMR and Analysis of

Experimental Data

The ﬁrst model of GMR was the semiclassical model

of Camley and Barnas (1989). This model is based on

the picture of free electrons scattered by a distri-

bution of spin-dependent scattering potentials that

Figure 9

Potential landscape seen by spin þand spin conduction

electrons in the parallel and antiparallel conﬁgurations.

The intrinsic potential is represented by a periodic array

of steps (Kronig–Penney-like potential); the bulk and

interface scattering potentials are represented by spikes.

Figure 10

Saturation MR vs. copper spacer layer thickness for several series of (Co (11 A

)/Cu (t

))

multilayers. The number of

Co/Cu periods, N, is 20 (ﬁlled circles) and 6 (open and ﬁlled squares). Data are shown for temperatures of (a) and (b)

295 K, and (c) and (d) 4.2 K. The N ¼6 data (in (a) and (c)) have been scaled by an empirical factor of 1.6 to make the

comparison with N ¼20 (after Parkin et al. 1993).

250

Giant Magnet oresistance

depends on the magnetic conﬁguration of the multi-

layer, and uses a Boltzmann equation formalism (see

Boltzmann Equation and Scattering Mechanisms).

Bulk scattering is taken into account by introducing

spin-dependent (spin-independent) relaxation times

in the magnetic (nonmagnetic) layers. Interface scat-

tering is described by introducing boundary condi-

tions at the interfaces, i.e., by assuming that a

proportion, T

, of the spin s electrons are transmit-

ted without scattering, while a proportion (1T

)is

diffusely scattered (a probability of specular reﬂec-

tion can also be introduced). The major success of

this model is that it predicts the thickness dependence

of GMR. It shows that the inﬂuence of the spin-

dependent scattering on the electron distribution

function does not extend further than about the

MFP inside the nonmagnetic layers, so that the re-

sistance becomes independent of the relative orienta-

tion of consecutive magnetic layers when they are

apart by more than the MFP.

Analytical expressions of GMR in the Camley–

Barnas model have been derived by Barthe

my and

Fert (1991) in the simple case where only interface

scattering is spin dependent. It is found that, in the

limit of thick layers, the MR ratio depends on the

nonmagnetic layer thickness, t

, as exp(t

where l

is the MFP in this layer. The variation as

1

at large values of the thickness, t

, of the magnetic

layers reﬂects the progressive dilution of the spin-

dependent interface scattering.

Figure 10 shows an example of experimental data

at 295 K for Co/Cu multilayers where the variation

as exp(t

) is observed at large thicknesses of

copper. At 4.2 K the exponential variation is not

observed and it is supposed that, with a larger l

low temperature, the exponential behavior is repelled

above the experimental thickness range.

The ﬁrst quantum mechanical model of GMR,

based on the Kubo formalism of transport, was

proposed by Levy et al. (1990). As in the model of

Camley and Barnas, it neglects the inﬂuence of the

intrinsic potential of the multilayer and describes

the scattering of free electrons by spin-dependent ex-

trinsic potentials. Figure 11 shows an example of

the application of the model to experimental data.

After 1993 the inﬂuence of the intrinsic potential of

the multilayer was progressively introduced into the

theoretical models and ab initio models were also

developed. For a general review of the models see

Barthe

my et al. (1999).

A general result is that the theory of CIP GMR

is far from being predictive. This comes from the

combination of the intrinsic and extrinsic contribu-

tions. Ab initio calculations can predict correctly the

inﬂuence of the intrinsic potential but the nature

and distribution of imperfections are generally too

poorly known to be introduced realistically into the

calculations.

5.2 Models of CPP GMR and Analysis of

Experimental Data

The advantage of CPP geometry for theory is that, in

the usual diffuse limit, the contribution from the in-

trinsic potential (steps in Fig. 9) can be expressed by

introducing spin-dependent interface resistances,

mðkÞ

¼ 2r



½1  ðþÞg, for the spin m and spin k chan-

nels. Boltzmann equation models have been devel-

oped to express CPP GMR as a function of the

interface parameters r



and g, and the parameters

describing the resistivity within the layers r

mðkÞ



½1  ðþÞb and r



for the magnetic and nonmag-

netic layers, respectively. An important point is the

treatment of the spin accumulation occurring when

the current is perpendicular to the interfaces. Valet

and Fert (1993) have shown that, owing to spin ac-

cumulation, the SDLs l

and l

in the magnetic

and nonmagnetic layers, respectively, become the

scaling lengths of the thickness dependence instead of

the MFP as in CIP geometry. As the SDL is much

larger than the MFP, this explains why CPP GMR

decreases more slowly as a function of the layer

thickness than CIP GMR.

Valet and Fert (VF) have derived general expres-

sions for CPP GMR as a function of the parameters

Figure 11

GMR ratio for Fe 30 A

/Cr multilayers vs. chromium

thickness. The experimental data (dots) are obtained at

4.2 K. The solid line is calculated within the quantum

model of (Levy et al . 1990) (after Gijs and Okada 1992).

251

Giant Magnet oresistance



, g, r



, b, r



, l

, and l

. In most experiments the

layer thicknesses are smaller than the SDL and, in

this so-called long SDL limit, the simple expressions

of a resistor series model (see Bass and Pratt 1999)

can be derived from the general expressions of the VF

model. For example, one obtains:

½ðR

 R

ÞR



1=2

¼ b

þ t



þ 2Ngr



ð6Þ

where t

and t

are the thicknesses of the magnetic

and nonmagnetic layers, N the number of period, and

L ¼N(t

þt

) is the total thickness. In multilayers

with t

¼t

and a given total thickness L, this gives

the linear variation of [(R

–R

]

1/2

as a function

of N, as illustrated in Fig. 12(a). Once r



, r



, and r



have been determined from the thickness dependence

of the resistance, the spin asymmetry coefﬁcients, b

and g, can be derived from, respectively, the slope and

the intercept with the vertical axis of the straight line

in Fig. 12(a). In the same way, for a given value of L

and t

, one obtains the linear variations as a function

of N with different slopes for different values of t

,as

illustrated in Fig. 12(b).

Measurements on extensive series of samples have

been analyzed in the above way to determine the pa-

rameters r



, g, r



, b, and r



for silver, copper, cobalt,

and NiFe layers, and Ag/Co, Cu/Co, Ag/NiFe and

Cu/NiFe interfaces (Bass and Pratt 1999). Once these

parameters have been determined, it turns out that

GMR can be predicted for any multilayer composed

of the same materials.

Simple expressions for CPP GMR can also be

found in the opposite limit, i.e., when the thickness is

much larger than the SDL.

For t

with t

, the VF model predicts

a reduction of GMR by a factor exp(t

). This

Figure 12

A{R

)[R

)–R

)]}

1/2

, where A is the sample

area, R

) the resistance in the virgin state (AP state),

and R

) the resistance at saturation (P state), against

N for Ag/Co and AgSn/Co multilayers with a ﬁxed total

thickness L ¼0.72 mm: (a) t

¼t

AgSn

and (b)

¼6 nm and 2 nm with t

varying. The resistivities of

the Ag and AgSn layers are 1 mO cm and 18 mO cm,

respectively (after Pratt et al. 1993).

Figure 13

{AR

)[AR

)AR

)]}

1/2

vs. the number of

bilayers, N, for Co/Cu, Co/CuMn (7%), and Co/CuPt

(6%), where A is the sample area and R

) and

) the resistances at zero ﬁeld and saturation ﬁeld,

respectively (after Yang et al. 1994). The dashed line

corresponds to the linear variation expected when the

SDL is much longer than the layer thicknesses. The solid

lines are ﬁts with the VF model with l

¼10 nm and

2.8 nm in CuPt and CuMn, respectively.

252

Giant Magnet oresistance

behavior is observed in Fig. 13 for multilayers in

which the SDL of the copper layers has been short-

ened by doping with manganese or platinum impu-

rities. The ﬁt with the VF expression (solid lines)

leads to the values of the SDL indicated in Fig. 13.

For pure nonmagnetic metals, the SDL is much larg-

er and can be determined only by measurements on

multilayered nanowires with very thick nonmagnetic

layers (Fert and Piraux 1999). For copper layers, one

obtains l

¼140 nm at low temperature.

In the limit t

with t

, the VF model

predicts:

ðR

 R

2pb

ð1  b

Þt

ð7Þ

where p is the proportion of antiparallel conﬁgura-

tion between consecutive magnetic layers. Experi-

mental results on the linear variation of the inverse of

GMR as a function of t

are shown in Fig. 14 and a

ﬁt with Eqn. (7) gives l

¼59718 nm for cobalt at

low temperature (45 nm at room temperature). The

SDL turns out to be even shorter in permalloy,

around 5 nm (Bass and Pratt 1999, Fert and Piraux

1999).

Figure 15

Schematic representation of the inverse GMR effect.

The signs þand are for spins s

¼þ

and 

respectively. The arrows represent the majority spin

direction in the magnetic layers. The electron trajectories

between two scatterings are represented by straight lines

and the scattering by abrupt changes in direction. When

and M

are made of two different magnetic materials

having opposite spin asymmetries, one of the spin

directions (e.g., the spin þdirection) is weakly scattered

everywhere in the antiparallel conﬁguration. This gives

rise to a short circuit effect and a smaller resistance in

this antiparallel conﬁguration, and consequently to an

inverse GMR effect.

Figure 14

Linear variation of the inverse of the MR vs. cobalt

thickness for Co (t

)/Cu (8 nm) multilayered nanowires

at T ¼77 K (ﬁlled symbols) and T ¼300 K (open

symbols). The circle refers to data obtained on a Co

(435 nm)/Cu (15 nm) sample (after Fert and Piraux 1999).

Figure 16

Variation of CPP GMR as a function of the thickness of

0.95

0.05

layers in NiCr=Cu=Co=Cu multilayers. The

competition between the bulk scattering (characterized

by b

NiCr

o0) and interface scattering (with g

NiCr=Cu

40)

gives rise to a compensation thickness t



NiCr

at which

they cancel each other (t



NiCr

is determined in the long

SDL limit by b

NiCr



NiCr



þ2g

NiCr=Cu



NiCr=Cu

¼ 0).

Below this compensation thickness the interface

scattering is predominant, the global spin asymmetry of

the

NiCr layer is positive as is that of the cobalt layer

(characterized by b

40 and g

Co=Cu

40), and the effect

is normal (insert (a)). Above t



NiCr

, the bulk scattering in

NiCr becomes predominant, so that the global

asymmetry of the NiCr layers is negative. Consequently,

with alternating spin asymmetries for the NiCr and

cobalt layers, the effect is the inverse (insert (b)) (after

Barthe

my et al. 1999).

253

Giant Magnet oresistance

6. Inverse GMR

Normal GMR, i.e., a smaller resistance in the parallel

conﬁguration, is obtained when the spin asymmetry

has the same sign in consecutive magnetic layers. This

is always the case for multilayers of the type (F/N)

But it is also possible to fabricate multilayers of the

type (F

/N/F

/N)

in which the ferromagnetic layers

and F

have opposite asymmetries, e.g., F

slow-

ing down more strongly the majority spin electrons

and F

the minority spin electrons. In these cases, it is

in the antiparallel conﬁguration that one of the spin

directions propagates easily everywhere and shorts

the current (see Fig. 15).

An example of experimental results is shown in

Fig. 16 for (NiCr/Cu/Co/Cu) multilayers in CPP ge-

ometry. The spin asymmetry coefﬁcients b

, g

Co/Cr

and g

NiCr/Cu

are positive while b

NiCr

is negative.

GMR becomes inverse for t

NiCr

42 nm when the

contribution from the bulk scattering (related to

NiCr

) exceeds the contribution from the interface

scattering (related to g

Co/Cr

) in the NiCr layers.

Figure 16 shows the variation of GMR as a function

of the thickness with a crossover from positive to

negative at about 2 nm. An example of an inverse

GMR curve can be seen in inset (b). Such experi-

ments on extensive series of (F

/N/F

/N)

multilayers

have been performed to determine not only the mag-

nitude but also the sign of the spin asymmetry co-

efﬁcients of a number of magnetic materials and

interfaces (Barthe

my et al. 1999).

7. GMR in Granular Systems

GMR effects have also been observed in granular

systems, i.e., magnetic clusters embedded in a non-

magnetic matrix (Xiao et al. 1992, Berkowitz et al.

1992). Figure 17 shows an example of a GMR curve

obtained in such systems. The resistivity drops

when the magnetic moments of the clusters are

aligned by an applied magnetic ﬁeld. The advantage

of granular materials is that they are not very sen-

sitive to the effect of pinholes. The drawbacks are

the large magnetic ﬁeld needed to overcome the an-

isotropy energy of nonspherical clusters and to

saturate the GMR (Fig. 17), and the crossover to

superparamagnetic behavior (Fig. 17(b)) above a

blocking temperature, which is generally below room

temperature.

With the aim of avoiding the propagation of pin-

hole effects and obtaining a large GMR effect in low

ﬁeld, some attempts have been made to prepare in-

termediate systems between multilayers and clusters.

Interesting results have been obtained starting from

multilayers and annealing them to create discontin-

uous layers of pancake-shaped clusters with low an-

isotropy energy. Another way to obtain intermediate

structures is to grow multilayers with very thin mag-

netic layers. Hybrid structures combining continuous

and discontinuous clustered magnetic layers have

also been grown. These kinds of structures lead to

large GMR ratios in very low ﬁelds (34% in 10 Oe

(796 Am

1

)) (Duvail et al. 1995).

Figure 17

(a) GMR curve obtained in granular alloys by codeposition of immiscible elements such as cobalt and silver. The Co

concentration is indicated on the curves. For concentrations above 28% cobalt, the clusters reach their percolation

threshold and the whole sample becomes ferromagnetic leading to the disappearance of the GMR effect. (b) Field-

cooled and zero ﬁeld-cooled curves measured in a Co

granular system. Above a blocking temperature around

100 K, the cobalt clusters are in a superparamagnetic state and a very large magnetic ﬁeld is needed to obtain the

GMR effect (after Berkowitz et al. 1992).

254

Giant Magnet oresistance

See also: Amorphous Intermetallic Alloys: Resis-

tivity; Giant Magnetoresistance: Metamagnetic

Transitions in Metallic Antiferromagnets; Magneto-

resistance: Magnetic and Nonmagnetic Intermetallics

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A. Barthe

my, A. Fert

UMR CNRS-Thomson CSF and Universite

Paris

Sud, Orsay, France

F. Nguyen Van Dau

UMR CNRS-Thomson CSF, Orsay, France

Giant Magnetoresistance: Metamagnetic

Transitions in Metallic Antiferromagnets

In intermetallic compounds with antiferromagnetic

(AF) ordering, a large change in the resistivity (mag-

netoresistance: MR) occurs within a magnetic ﬁeld,

associated with a metamagnetic transition (MT). The

mechanisms responsible are discussed here in relation

to several examples.

The MR ratio is deﬁned as:

DrðBÞ

rð0Þ

rðBÞrð0Þ

rð0Þ

ð1Þ

where r(B) is the resistivity under a magnetic ﬁeld H

(B ¼m

H).

The term ‘‘giant magnetoresistance’’ (GMR) was

ﬁrst used for the large resistance decrease in magnetic

multilayers (MML) associated with the ﬁeld-induced

255

Giant Magnetoresistance: Metamagnetic Transitions in Metallic Antiferromagnets

change of interlayer coupling from AF to ferromag-

netic (F) (Baibich et al. 1988). The spin-dependent

scattering, namely the difference in the relaxation

times between the up- and down-spin electrons, is

now recognized to be the origin.

1. GMR in Layered Antiferromagnets

UNiGa (hexagonal ZrNiAl-type structure) was the

ﬁrst intermetallic compound discussed in relation

with the GMR in MML (Sechovsky et al. 1991). The

ground state AF structure in UNiGa is characterized

by the c-axis stacking of U-moments mmkmkk and

oriented along the c-axis ferromagnetically ordered

within the basal plane sheet. The magnetic ﬁeld ap-

plied along c induces a MT at B

B0.8T (at 4.2K)

leading to F stacking, and accompanied by a negative

GMR. Figure 1 compares the ﬁeld dependencies of

MR and magnetization measured on the same crys-

tal. The GMR is largely anisotropic: Dr/r

¼90% for

B:i and 56% for B>i. Besides the anisotropy, the

geometrical resemblance to the MLL initially led to

the conclusion that the GMR in UNiGa has the same

origin as MML: the spin-dependent scattering

(Sechovsky et al . 1993). However, the inherent as-

sumptions for the GMR in MML, the dominance of

conduction electron scattering at interlayer bounda-

ries, and the spin-split density of states for each

U-monolayer, are not appropriate for UNiGa.

2. Mechanisms of GMR–Fermi Surface

Reconstruction

The GMR in UNiGa might be more naturally as-

cribed to a Fermi surface (FS) reconstruction con-

nected with the disappearance of the superzone gap

across the MT. Figure 2 shows a simpliﬁed model for

MT, across which the unit cell size along c changes

from 2c to c. As a result, the superzone gap at 7p/2c

disappears in the wave vector (k-) space. Even from

such a simple model, one can infer the reduction of

resistance across the MT. The electrical conductivity

tensor s

along the i-axis is given as an integral over

FS as (Dugdale 1977)

tðkÞv

ðkÞ

7v7

ð2Þ

where t(k) and v

(k) are the relaxation time and the

i-component of velocity, respectively, for a conduc-

tion electron with a wave vector k. The change in

conduction electron scattering is reﬂected in t(k),

while those in FS appear both in the FS area through

and in the velocity v(k).

Generally, the AF superzone gap reduces the FS

area (Elliott and Wedgwood 1963), leading to an

isotropic reduction of resistance, as expected from

Eqn. (2). In addition, the AF superzone gap reduces

the velocity component normal to the Brillouin zone

boundary as shown in Fig. 2. Thus, the velocity fac-

tor reduces the conductivity along the c-axis selec-

tively across the MT, which explains the anisotropic

MR in Fig. 1. Antonov et al. (1996) have succeeded in

ascribing the GMR in UNiGa to the FS reconstruc-

tion semiquantitatively by ﬁrst-principle calculations

based on the density-functional band structure. This

scenario has been experimentally conﬁrmed both by

the specific heat and the Hall effect measurements

(Aoki et al. 1997, Kobayashi et al. 1996).

SmMn

(van Dover et al. 1993) and FeRh

(Schinkel et al. 1974), which also posses the AF

stacked two-dimensional ferromagnetic planes, ex-

hibit a negative MR accompanying the MT as well.

3. GMR in Nonlayered Antiferromagnets

The AF stacking of two-dimensional ferromagnetic

planes is not a necessary condition for the GMR. For

more general AF structures, numerous intermetallic

compounds have been reported to exhibit a large MR

correlated with the MT. NdGa

¼9.5K) with the

AlB

type hexagonal structure and PrFe

¼6.2K) with the cubic ﬁlled Skutterudide struc-

ture may serve as examples.

Figure 3 shows the transverse MR and magnetiza-

tion measured on a NdGa

single crystal at 4.2K be-

low T

where the magnetic structure is close to the

antiphase type. The magnetization exhibits a multistep

Figure 1

Field dependence of (a) magnetization and (b)

magnetoresistance in UNiGa.

256

Giant Magnet oresistance: Metamagnetic Transitions in Metallic Antiferromagnets

metamagnetic behavior, which is well reﬂected in the

MR curve. Even in the intermediate magnetic conﬁg-

uration, the MR is always negative, which suggests the

existence of conduction electron scattering in zero ﬁeld

state resulting from the low energy excitation, as was

inferred from the specific heat measurement (Aoki

et al. 1997).

Figure 4 shows the ﬁeld dependence of the MR in

PrFe

along with the magnetization above and

below the Ne

el temperature (T

) of 6.5K. For this

material, the FS reconstruction at T

can be clearly

seen as a jump in the temperature dependence of the

resistivity in zero ﬁeld. This was also manifested in

the temperature dependence of the Hall coefﬁcient

(Sato et al. 2000). A notable feature in Fig. 4 is

the initial increase of resistivity in ﬁelds below B

Yamada and Takada (1973) have explained such an

effect by the critically enhanced conduction electron

scattering on magnons. They calculated the MR due

to electron-spin scattering in a simplest uniaxial AF

metal considering the s–d interaction. For the ﬁeld

parallel to the easy axis, they found a positive MR

below B

, which results from the increase in the total

number of excited magnons. More intuitively, the

enhancement of spin ﬂuctuations in the antiparallel

sublattice b in Fig. 2 overcomes the suppression of

spin ﬂuctuations in the parallel sublattice a. On the

Figure 2

Simplest model for the MT. Top and bottom ﬁgures correspond to the real and k-space geometry. In zero ﬁeld (left),

two-dimensional ferromagnetic sheets form AF structure made of parallel (a) and antiparallel (b) sublattices. The

superzone gap formed in AF state disappears across the MT.

Figure 3

Field dependence of (a) magnetization and (b) MR ratio

in NdGa

257

Giant Magnetoresistance: Metamagnetic Transitions in Metallic Antiferromagnets

other hand, Baranov et al. (1993) ascribed the anom-

alous increase of the resistivity to the conduction

electron scattering at interface boundaries separating

the phases with different magnetic structures.

To summarize, in the process of the MT from an

AF state, the GMR has been observed in variety of

intermetallic compounds and ordered alloys. The

disappearance of the AF superzone gap across the

MT plays the main role in the GMR in many cases.

Additional mechanisms, such as the spin ﬂuctuation

scattering, the domain wall scattering, etc., should

also be taken into account in order to explain the

overall behavior including the temperature variation

of the MR. Especially for itinerant magnetic electron

systems such as UNiGa, the splitting of the up- and

down-spin sub-bands might lead to a further change

of FS in the high-ﬁeld ferromagnetic state.

See also : Amorphous Intermetallic Alloys: Resistiv-

ity; Giant Magnetoresistance; Magnetoresistance,

Anisotropic; Magnetoresistance: Magnetic and Non-

magnetic Intermetallic

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Tokyo Metropolitan University, Japan

Giant Magnetostrictive Materials

The magnetostrictive materials exhibit a strain caused

by the orientation of the magnetic moment when

exposed to a magnetic ﬁeld (see Magnetostrictive

Materials). A new class of magnetostrictive materials

discovered by Ullakko and co-workers (1995, 1996) is

Figure 4

Field dependence of (a) magnetization and (b) MR in

PrFe

258

Giant Magnet ostrictive Materials

called magnetic shape memory (MSM) alloys or

ferromagnetic shape memory alloy (FSMA) materi-

als. The thermoelastic martensitic phase transforma-

tion in MSM materials produces a low-symmetry

phase with a large magnetocrystalline anisotropy and

highly mobile twin boundaries between the variants.

MSM materials exhibit giant magnetic-ﬁeld-induced

strain (MFIS) based on the rearrangement of the

crystallographic domains (twin variants). In the mag-

netic ﬁeld, those martensite variants having the easy

axis of magnetization along the ﬁeld, start to grow

due to twin boundary motion and become dominant.

This process lowers the magnetization energy. The

MFIS of MSM materials is unique since it produces a

large strain with rather high frequencies without a

change in the external temperature. Consequently,

MSM materials are potentially important for actua-

tor and sensor applications (Tellinen et al. 2002).

The currently best working MSM materials are the

near stoichiometric Ni

MnGa Heusler alloys. Their

structural, magnetic, and mechanical properties are

highly sensitive to the chemical composition and

temperature. These have been studied in detail by

researchers in several countries (Vasil’ev et al. 2003,

derberg et al. 2004). Depending on the martensite

crystal structure (5 M or 7 M), one can obtain 6% or

10% strain response in a magnetic ﬁeld less than

800 kA m

1

(Murray et al. 2000, Sozinov et al. 2003),

with the optimum frequencies up to 300–500 Hz

(O’Handley and Allen 2001) in the temperature range

150–333 K (Heczko and Straka 2003) and at ambient

temperature the MSM fatigue life can be more than

50 10

shape change cycles.

The schematic behavior of an MSM material is

shown in Fig. 1. In the Ni–Mn–Ga 5 M and 7 M

martensitic structures, the c-axis is the shortest cry-

stallographic axis and, simultaneously, the easy axis

of magnetization. When the material is exposed to a

magnetic ﬁeld, the variants having c-axis along the

ﬁeld become dominant and, consequently, the mate-

rial contracts in the direction of the applied ﬁeld.

Since this MFIS remains after removing the magnetic

ﬁeld (Fig. 1 (1)), the actuation movement is only ob-

tained by turning the magnetic ﬁeld perpendicular to

its original orientation (e.g., applying a rotating mag-

netic ﬁeld, Fig. 1 (2)) or with an external spring-back

load (Fig. 1 (3)). Consequently, with this way, it is

possible to change one single-variant structure to an-

other and to obtain cycling of the giant MFIS.

Figure 2 represents the behavior of the MSM ma-

terial under compressive stress in a constant trans-

versal magnetic ﬁeld (H). The opposite stress induced

by the magnetic ﬁeld (Ds

mag

) is growing with in-

creasing H and it saturates at Ds

mag

¼(1c/a)

1

where K

is the magnetocrystalline anisotropy energy

density of martensite (see insert in Fig. 2). Conse-

quently, the giant MFIS is limited to the materials

with rather low twinning stress ( s

) as the require-

ment for MFIS is Ds

mag

(Likhachev et al. 2001).

The mechanical behavior of an MSM element in a

constant transversal magnetic ﬁeld (in Fig. 2) resem-

bles the conventional superelastic (SE) behavior of

the austenitic alloys: the large strain related to the

Figure 1

MSM element in (1) unidirectional, (2) rotating

magnetic ﬁeld, and (3) in the actuator. a, b and c are

crystallographic axes of martensite (co a; b), c-axis is

easy axis of magnetization.

Figure 2

Compression of MSM element in constant transversal

magnetic ﬁeld H ¼800 kA m

1

. s

is twinning stress

and Ds

mag

the magnetic stress. Inset shows inﬂuence of

magnetic ﬁeld on Ds

mag

259

Giant Magnet ostrictive Materials