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

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5. Summary

The inﬂuence of an external magnetic ﬁeld on the

resistivity is the subject of this article. The effect of a

ﬁeld on the electrical resistivity can be considered as

consisting of two parts. The ﬁrst originates from the

inﬂuence of the magnetic ﬁeld on the conduction

electron trajectories owing to the Lorentz force,

which deﬂects the electrons on their way through

the specimen. This mechanism always increases the

resistivity with increasing ﬁeld. The other part is ow-

ing to the inﬂuence of the external magnetic ﬁeld on

the magnetic state and thus on the spin-dependent

scattering mechanism. The effect is that in the para-

magnetic and the ferromagnetic cases the resistivity

decreases with increasing ﬁeld, whereas in the anti-

ferromagnetic state an increase of the magnetoresist-

ance is in general expected.

Acknowledgments

The author wishes to express his thanks to his col-

leagues H. Nowotny and K. Hense for many stim-

ulating discussions.

See also : Amorphous Intermetallic Alloys: Resistiv-

ity; Elemental Rare Earths: Magnetic Structure and

Resistance, Correlation of; Giant Magnetoresistance;

Giant Magnetoresistance: Metamagnetic Transitions

in Metallic Antiferromagnetics; Magnetoresistance,

Anisotropic; Magnetism in Solids: General Introduc-

tion; Localized 4f and 5f Moments: Magnetism; Al-

loys of 4f (R) and 3d (T) Elements: Magnetism; Rare

Earth Intermetallics: Thermopower of Cerium, Sa-

marium and Europium Compounds

Bibliography

Blatt F L 1968 Physics of Electronic Conduction in Solids.

McGraw-Hill, New York

Butcher P N 1973 Basic electron transport theory. In: Electrons

in Crystalline Solids. IAEA, Vienna, pp. 103–65

Campbell I A, Fert A 1982 Transport properties of ferromag-

nets. In: Wohlfarth E P (ed.) Ferromagnetic Materials. North

Holland, Amsterdam, Vol. 3, pp. 747–804

Elliott R J, Wedgwood F A 1963 Theory of the resistivity of the

rare earth metals. Proc. Phys. Soc. 81, 846–55

Gratz E, Maikis M, Bauer E, Nowotny H 1995 Magneto-

resistance in magnetic and nonmagnetic rare earth com-

pounds. J. Magn. Magn. Mater. 140–144, 111–2

Gratz E, Nowotny H, Hense K, Enser J, Markossian S, Tanaka

T 2001 Magnetoresistance in ferro-, antiferro- and non-

magnetic rare earth compounds. To be published

Kaneko T, Yoshida H, Ohashi M, Abe S 1987 Magnetic struc-

ture of the intermetallic compound DyAg. J. Magn. Magn.

Mater. 70, 277–8

Kasuya T 1968 Effects of s–d interaction on transport phe-

nomena. Prog. Theor. Phys. 22, 227–46

Lacueva G, Levy P M, Fert A 1982 Uniﬁed approach to some

transport and EPR properties of noble metals with rare-earth

impurities. Phys. Rev. B 26 (3), 1099–124

Yamada H, Takada S 1973a Magnetoresistance of antiferro-

magnetic metals due to s–d interaction. J. Phys. Soc. Jpn. 34,

51–7

Yamada H, Takada S 1973b Magnetoresistance due to elec-

tron-spin scattering in antiferromagnetic metals at low tem-

peratures. Prog. Theor. Phys. 45, 1401–19

E. Gratz

Technische Universita

t Wien, Vienna, Austria

Magnetoresistive Heads: Physical

Phenomena

Electrical resistivity is a powerful probe of the elec-

tronic and physical structure of a material. This sen-

sitivity has resulted in electrical resistance commonly

being used to sense indirectly other material proper-

ties such as temperature or defect concentration.

Electrical resistance can also vary with the magnetic

state of the material, magnetoresistance, stimulating

the development of magnetoresistive magnetic ﬁeld

sensors that are not compromised by the inherent

speed dependence of inductive ﬁeld sensing. The an-

isotropic magnetoresistance (AMR) (McGuire and

Potter 1975) of bulk ferromagnetic materials has,

Figure 9

Temperature variation of the transverse

magnetoresistance of the cubic antiferromagnetic DyAg

compound. Dr

/r vs. T shows the characteristic

increase at the onset of the antiferromagnetic ordering.

Inset shows the temperature variation of w

(T/T

)

determined from experiment for DyAg. It exempliﬁes

the correspondence with the theoretical w

function

shown in Fig. 8. The agreement is less good for the

temperature region with the sinusoidally modulated spin

conﬁguration (T

oToT

830

Magnetoresistive Heads: Physical Phenomena

until recently, provided the greatest potential for

magnetic ﬁeld sensors.

The development of the technology to create mag-

netic thin-ﬁlm structures with layer thicknesses of the

order of the electron mean free path led to the dis-

covery of giant magnetoresistance (GMR) (Baibich

et al. 1988) that promised an increase of one to two

orders of magnitude in ﬁeld sensitivity (see Giant

Magnetoresistance). GMR sensors are now to be

found as the ﬁeld-sensing element in the read heads of

commercial hard disks (Toigo 2000). The origins of

GMR lie in the spin-dependent scattering experienced

by an electron traveling through the material that

results from the spin-polarized electronic band struc-

ture of a ferromagnetic material. Both intrinsic and

extrinsic factors affect the magnitude and ﬁeld sen-

sitivity of the GMR.

1. AMR

AMR was ﬁrst discovered by Lord Kelvin. Due to

spin–orbit coupling, the orientation of the electron

orbitals in a ferromagnetic material is determined by

the direction of the magnetic ﬁeld (see Magnetore-

sistance, Anisotropic). This results in a greater scat-

tering cross-section for a transport electron when the

direction of the current is parallel to the applied ﬁeld

(McGuire and Potter 1975) (see Fig. 1):

r ¼ r

þ Dr

AMR

cos

y ð1Þ

where y is the angle between the applied ﬁeld and the

current. This effect is greatest in materials with large

spin–orbit coupling such as Ni

where magne-

toresistance values of 4% are possible.

2. Origin of Spin-dependent Transport

The band structure of the free electron like s electrons

in ferromagnetic materials is not spin polarized; how-

ever, they have the highest mobility and would be

expected to be responsible for carrying the majority

of the current. However, the 3d electrons have the

greatest density of states at the Fermi surface and

their band structure is spin split. The probability of

electron scattering is proportional to the density of

states at the Fermi surface resulting in a probability

of s electrons scattering into empty states in the spin

split d band and hence spin-dependent scattering. The

spin asymmetry parameter is often represented by

a ¼ R

ð2Þ

A schematic diagram of the spin split band struc-

ture of a strong ferromagnet such as cobalt and nickel

is shown in Fig. 2. In the example shown, the higher

density of states at the Fermi level for the spin down

band results in a higher probability of scattering for

this band and hence a value of a less than unity. Iron

is only a weak ferromagnet; however, its band struc-

ture is readily modiﬁed by distortion of its local elec-

tronic environment, for example at a surface or

interface. First principles band structure calculations

are often necessary to determine the degree of spin-

dependent scattering, especially for alloys such as

3. GMR

Mott in 1936 determined that the spin quantum

number is usually conserved during scattering pro-

cesses allowing the transport of spin up and spin

down electrons to be treated as two parallel spin

currents. Taking into account the spin-dependent

scattering described in Sect. 2, consider the schematic

model shown in Fig. 3.

Figure 1

Orientation dependence of AMR.

Figure 2

Schematic electronic band structure for a strong

ferromagnet, e.g., cobalt.

831

Magnetoresistive Heads: Physical Phenomena

The magnetic elements represent regions of satu-

rated magnetization whose relative orientation can be

altered as indicated. The resistances of the spin up and

spin down channels are modeled as resistors, a larger

resistor representing a larger resistance, and are

shown acting in parallel in accordance with the Mott

two-current model. A current entering the device will

contain both spin up and spin down electrons. In case

(a) spin up electrons will remain majority electrons

throughout their passage and will only be weakly

scattered. Spin down electrons, however, will remain

minority electrons throughout their passage and will

be strongly scattered throughout. The overall resist-

ance in case (a) will therefore be dominated by the

low-resistance spin up channel and the current will be

spin polarized in the up direction as it emerges.

By contrast, in case (b) where the magnetic regions

are misaligned, both the spin up and spin down elec-

trons will spend some time as both majority and mi-

nority electrons resulting in a high resistance in both

channels. If the device can be switched between these

two magnetic states a change in resistance will be

observed. The magnetoresistance due to this mecha-

nism can be represented by

 R

ð1  aÞ

ð3Þ

where R

) is the resistance in the antiparallel

(parallel) conﬁguration, and is called GMR. The name

refers to the mechanism responsible for the resistance

change rather than to the size of the effect itself.

Bulk magnetic material is normally divided up into

regions of saturated magnetization called magnetic

domains; however, within the domain wall separat-

ing domains of different magnetization direction, the

local direction of the magnetic moment rotates. As

the spin-polarized electrical current travels through

the domain wall its spin may be able to rotate about

the local direction of magnetization thus quenching

any GMR effect (Gregg et al. 1996). The thinner the

domain wall, the more rapid the rotation of the local

moment will be and the lower the probability that the

electron spin will be able to precess with the local

moment. Therefore, to increase the GMR effect the

domain wall should be made as thin as possible. By

separating adjacent magnetic domains by a nonmag-

netic material the domain wall is effectively removed.

This is best achieved in a magnetic multilayer in

which the layer thicknesses can be reduced to a few

atomic layers.

This was ﬁrst achieved by Baibich et al. (1998) and

Binasch et al. (1989) in Fe/Cr multilayers; the original

data showing GMR for the ﬁrst time is reproduced in

Fig. 4.

Since this initial discovery the subsequent under-

standing of the underlying mechanisms has enabled

GMR of 220% to be achieved at 4.2 K in Fe/Cr

multilayers (Schad et al. 1994), and at room temper-

ature values of 65% have been realized for Co/Cu

multilayers (Parkin et al. 1991).

Given that multilayering magnetic and nonmag-

netic materials creates a good system for spin-

dependent scattering, the choice of materials and

the microstructure of the multilayer will determine

the maximum value of GMR. However, for effective

application as a magnetic ﬁeld sensor this magneto-

resistance must also occur for small changes in ap-

plied ﬁeld. The mechanism by which the multilayer

switches between the two magnetic states of parallel

and antiparallel alignment of adjacent magnetic lay-

ers is therefore crucial in GMR design. Despite their

lower magnetoresistance, AMR devices do in fact

have quite a good ﬁeld sensitivity, as a thin ﬁlm of

, for example, can be switched in ﬁelds of

only a few tenths of a millitesla.

Figure 3

Spin-dependent scattering leading to GMR in a trilayer represented using the resistor model.

832

Magnetoresistive Heads: Physical Phenomena

In the schematic diagram in Fig. 2 the current is

shown traveling perpendicular to the layers in the

current perpendicular to the plane (CPP) geometry.

In this case the resistances of subsequent layers can

be added in series and this is the most straightforward

type of structure to model theoretically. However,

owing to the large cross-sectional area to length as-

pect ratio, the CPP resistance is extremely small

making practical realization more difﬁcult. Super-

conducting contact pads are often used (Pratt et al.

1991) or pillars are formed by etching (Gijs et al.

1993) or electrodeposition (Piraux et al. 1994) to re-

duce the cross-section and thereby increase the meas-

urement resistance.

As the electron mean free path in a typical metal

multilayer is typically much longer that the thick-

nesses of the individual layers, it is possible to meas-

ure GMR with the current in the plane (CIP)

geometry in which the current travels parallel to the

layers. In this case the resistances of the layers are

added in parallel for each spin current channel. CIP

magnetoresistance is practically much easier to meas-

ure, as the resistance values are typically a few ohms.

There are now many designs of nanostructures that

enable the relative alignment of the magnetic com-

ponents to be controlled by the application of a

magnetic ﬁeld—some of which are described below.

They enable the variation of magnetic ﬁeld to be

controlled according to the following angular-

dependent equation:

RðyÞ¼Rðy ¼ 0ÞþDR

GMR

1  cosy

ð4Þ

where y is the angle between the magnetization di-

rections of adjacent layers.

3.1 Exchange Coupled Multilayers

GMR was discovered in an Fe/Cr magnetic multi-

layer whose antiparallel alignment of magnetic layers

was the result of interlayer exchange coupling. This

coupling is due to spin-dependent reﬂection coefﬁ-

cients at interfaces setting up a spin density wave in

the interlayer. The thickness of the interlayer deter-

mines at which point of the spin density wave the next

magnetic layer is situated resulting in either anti-

ferromagnetic (antiparallel) alignment or ferromag-

netic (parallel alignment) coupling between the

magnetic layers (Gru

nberg et al. 1986). This cou-

pling is much stronger than that arising from dipolar

interactions. The application of a magnetic ﬁeld to an

antiferromagnetically coupled multilayer will eventu-

ally overcome the coupling energy and align the

magnetic layers in the direction of the magnetic ﬁeld.

The relative alignment of the magnetic layers in the

multilayer can therefore be altered by the application

of a magnetic ﬁeld resulting in GMR. The disadvan-

tage of an exchange coupled multilayer is that for

strong coupling a large ﬁeld is often required to

overcome the exchange coupling resulting in poor

ﬁeld sensitivity of the GMR. In addition, the mag-

nitude and sign of the exchange coupling is very sen-

sitive to interlayer thickness; the coupling typically

cycles between ferromagnetic and antiferromag-

netic with a period of only 1.1–1.2 nm. This makes

Figure 4

GMR in Fe/Cr multilayers (after Baibich et al. 1988).

833

Magnetoresistive Heads: Physical Phenomena

the reproducible manufacture of such systems very

demanding and the problem of interdiffusion at the

magnetic/nonmagnetic interface very significant (see

Multilayers: Interlayer Coupling).

3.2 Spin Valves

In a spin valve (Dieny et al . 1991a), the exchange

coupling between the magnetic layers is suppressed

by separating them with a thick nonmagnetic inter-

layer. One of the magnetic layers is then pinned by,

for example, exchange biasing with an antiferromag-

net, e.g., FeMn. The other magnetic layer is left free

to magnetically switch in a small magnetic ﬁeld. The

most sensitive GMR materials have been made in this

way producing values in the region of 18% kAm

1

(see Magnetic Recording Systems: Spin Valves). Ex-

change biased multilayers suffer from the relatively

low Curie temperature of the antiferromagnet. This

limitation has been overcome by replacing the pin-

ning antiferromagnet by an antiferromagnetically

coupled multilayer such as Co/Ru (van den Berg

et al. 1996).

3.3 Heterogeneous Alloys

Layering of the materials is not a requirement for

GMR, but the dimensions of the magnetic and non-

magnetic regimes must be on the nanoscale. For ex-

ample, granular materials will also exhibit GMR

(Berkowitz et al. 1992, Xiao et al. 1992). Such systems

are most easily realized using immiscible magnetic

and nonmagnetic materials such as CoAg and can

readily be prepared by coevaporation, cosputtering,

or even mechanical alloying to produce a bulk ma-

terial with a nanoscale microstructure. The highest

resistance is observed when there is the most mag-

netic disorder at the coercive ﬁeld. In general, the

ﬁeld sensitivity is not high owing to the relatively high

ﬁelds often needed to rotate the magnetic moment of

small magnetic particles. As the GMR is very sensi-

tive to the moment–moment correlation of neighbor-

ing particles, it can be used to probe the magnetic

state of the sample (Gregg et al. 1996).

3.4 Tunneling Magnetoresistance

If the nonmagnetic layer is replaced by a thin insu-

lator such as Al

, the quantum mechanical tunnel-

ing probability through the insulator will also be spin

dependent owing to the number of available states to

tunnel into leading to a tunnel magnetoresistance

(TMR) (Juliere 1975).

4. Critical Length Scales

The two critical length scales in giant magnetoresis-

tive structures are the electron mean free path and the

spin diffusion length. The spin diffusion length is the

length over which the polarization of the electron

current is destroyed owing to spin ﬂip scattering. It is

related to the momentum mean free path by the fol-

lowing:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð5Þ

where v

is the Fermi velocity, l the momentum mean

free path, and t

the spin ﬂip time.

Typical values for a GMR multilayer are 10 nm for

the mean free path and 100–1000 nm for the spin

diffusion length in the nonmagnetic material. For a

CIP structure, the mean free path is usually the crit-

ical parameter as it will control the number of layers

an electron will typically encounter in its journey

through the multilayer. In a CPP structure, the elec-

tron will always sample all the layers in the structure

and it is therefore the spin diffusion length that is

usually the critical length scale as it determines the

degree of polarization of the current as each new in-

terface is reached. In addition, the spin diffusion

length in the magnetic layer will affect the degree of

polarization of the electron current. Heterogeneous

alloys exhibit both CIP and CPP character with the

resistances of the magnetic particles adding in both

series and parallel. Both length scales are therefore

important in these materials.

5. Sources of Electron Scattering

All scattering will reduce the mean free path of the

material. All spin-independent scattering, such as

scattering in the nonmagnetic layer, will therefore

reduce the GMR. This scattering may be the result of

the microstructure such as vacancies, defects, inter-

diffusion, and roughness at the surface or interface or

stress and can often be improved by the careful

choice of underlayer materials. In a thin ﬁlm there

will also be scattering from the interface with the

substrate and the capping layer. Scattering in the

ferromagnetic layer or at the interface may be spin

dependent and may therefore make a positive con-

tribution to the GMR.

The potential of the scattering process to be spin

dependent is interdependent on intrinsic properties

such as the band structure, the composition of the

multilayer, and its microstructure. Scattering can be

spin dependent if there is a matching of the atomic

potentials for one spin direction and not in the other;

for example, in the Fe/Cr system there is good band

matching for the spin down band, but poor matching

for the spin up band leading to spin-dependent scat-

tering and GMR. The reverse situation is true for

nickel or cobalt at an interface with copper where the

spin up band of the ferromagnet is full. Poor struc-

tural matching between materials at an interface re-

sults in a high dislocation density with high

consequent scattering. Note that the highest values

834

Magnetoresistive Heads: Physical Phenomena

of GMR at room temperature are obtained for the

Co/Cu system for which there is good matching of

both the electronic and lattice structure.

Additional complications arise for atomic mixing at

the interface where the magnetic moment of the alloy

thus produced is different to the bulk material, e.g.,

Ni/Cu (Nicholson et al. 1994). This has additional

consequences for exchange coupled material where the

interdiffusion will also affect the exchange coupling

and hence the degree of misalignment of the adjacent

magnetic layers. An effective way to beneﬁt from, say,

the soft magnetic properties of Ni

, avoid the

problems of interdiffusion at the Ni

/Cu inter-

face, and exploit the effective spin-dependent scattering

at the well-matched Co/Cu interface is to dust the

/Cu interface with a thin layer of cobalt

(Parkin 1993). In general, the band structure at the

interface is likely to differ from the bulk making it

difﬁcult to predict the performance of a particular in-

terface (Coehoorn 1993, Itoh et al. 1993, Nicholson

et al. 1994).

6. Effect of the Layer Thicknesses

The function of the nonmagnetic layer is to separate

the magnetic layers sufﬁciently to allow independent

switching in a spin valve material and in an exchange

coupled material to control the interlayer coupling

and thereby the degree of magnetic layer misalign-

ment. Increasing the thickness of the nonmagnetic

layer beyond what is required to achieve these func-

tions will result in a degradation of the GMR. Typ-

ically the nonmagnetic layer has a higher conductivity

than the ferromagnet, so in a CIP structure it acts as

an effective electrical shunt. Some materials com-

monly used as nonmagnetic layers such as gold also

have a high degree of spin ﬂip scattering resulting

from large spin–orbit coupling. This will reduce the

spin diffusion length and hence be particularly de-

grading to the GMR in CPP structures. There is,

however, a risk for both CIP and CPP structures in

making the nonmagnetic layer too thin owing to the

potential for the formation of structural pinholes that

will result in direct magnetic coupling between the

magnetic layers destroying the magnetic misalign-

ment and with it the GMR (Dieny et al. 1991b).

The magnetic layer must be thick enough to effec-

tively spin polarize the current; if it is too thin there

will be a high degree of diffuse scattering at the in-

terfaces reducing the mean free path considerably.

The GMR will increase with the number of layers

until in a CIP structure the mean free path is exceed-

ed; there is therefore also an electrical shunting effect

associated with the magnetic layer (Dieny et al. 1992).

7. Temperature Dependence

The temperature dependence of the GMR is neces-

sarily connected to the Curie temperature of the

magnetic material, and the formation of magnons

increases the amount of spin ﬂip scattering as the

temperature is increased. The temperature at which

the GMR is reduced to zero is lower than the Curie

temperature, but related to it. The increased conduc-

tivity of the whole magnetic structure at low temper-

atures results in a reduction of spin-independent

scattering, and an increase in the electron mean free

path and consequently in an increase in the GMR

ratio. For spin valves it is found that the GMR de-

creases linearly with increased temperature (Dieny

et al. 1992).

8. Theories of GMR

A model of GMR needs to include spin-dependent

scattering probabilities in the bulk, the outer bound-

aries, and at the interfaces within the structure. As

the origin of the spin-dependent scattering derives

from the spin splitting of the d band, a truly classical

model is incapable of describing the GMR. However,

a great deal of physical insight can be gained by in-

corporating a quantum mechanically derived spin

dependence into the scattering parameters of classical

theories to form semiclassical models.

The ﬁrst semiclassical model was that of Camley

and Barnas (1989) who used the Boltzmann transport

equation. The Fuchs and Sondheimer formulation

can be used to take into account the effect of the ﬁnite

thickness of the thin ﬁlm. The spin-dependant prob-

ability of specular reﬂection at the outer boundary

must be considered and at the interface: transmission,

specular reﬂection, and diffuse scattering must all

be taken into account in deriving spin-dependent

transmission coefﬁcients. These models lead to the

conclusion that in Ni

-based materials spin-de-

pendent scattering within the ferromagnetic layer is

dominant, in contrast to cobalt-based multilayers in

which the interface is found to dominate. In both

cases the detailed microstructure plays a significant

role and can be the dominant factor.

Valet and Fert (1993) describe spin-dependent

transport using the idea of spin accumulation at the

internal interfaces and use a semiclassical formalism

of the Boltzmann equation with the addition of

spin-dependent electrochemical potentials. These

models work well in CPP structures when the spin

diffusion lengths in both the ferromagnetic and non-

magnetic layers are much greater than the thicknesses

of the individual layers. Pratt et al . (1997) have

incorporated the spin diffusion length of the ferro-

magnetic material into a similar model.

Quantum mechanical theories take into account

ﬁnite size effects on the electronic structure and the

transport owing to the presence of a potential barrier,

but have been less successful in predicting realistic

values of the GMR. Realistic band structures have

been successfully incorporated into these models,

835

Magnetoresistive Heads: Physical Phenomena

most notably by Tsymbal and Pettifor (1996), who

considered hybridization of the sp and d electrons.

They considered scattering from spin-independent

disorder introduced using the Anderson model, using

the Kubo formula. The degree of disorder can be

altered as a parameter to obtain typical resistivities

for thin-ﬁlm structures and, for the ﬁrst time, realistic

values obtained for the GMR.

9. Summary and Future Developments

Although the underlying mechanisms of GMR are

now understood, accurate predictions are still difﬁ-

cult owing to the significant effect minor changes in

electronic band structure and microstructure can

have on the spin-dependent scattering. GMR sensors

have successfully found a major market for applica-

tions most notably as read heads in the data storage

industry (Toigo 2000) (see Magnetic Recording De-

vices: Inductive Heads, Properties) and increasingly in

a wider range of applications. The latest develop-

ments are in magnetic random access memory

(MRAM) (Pohm et al. 1994) (see Random Access

Memories: Magnetic) and hybrid spin electronic de-

vices (Gregg et al. 1997, Hammar et al. 1999) (see

Magnetic Recording Systems: Spin Electronics). More

detailed reviews of GMR can be found in the liter-

ature (e.g., Dieny 1994, Gijs and Bauer 1997).

See also: Colossal Magnetoresistance Effects: The

Case of Charge-ordered Pr

0.5

MnO

Manganite

Thin Films; Magnetoresistance in Transition Metal

Oxides

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S. M. Thompson

University of York, York, UK

Magnetostrictive Materials

Magnetostriction is the term given to the mechanical

deformation induced in a solid substance by the

modiﬁcation of its magnetic state. Inverse magneto-

striction refers to the modiﬁcation of the magnetic

state by an applied stress.

These phenomena are significant only in substances

which are magnetically ordered or at least close to

the onset of a magnetic order, and have practical im-

portance only in those materials which exhibit a spon-

taneous magnetization, i.e., ferro- or ferrimagnetic

materials. Only the latter will be considered in this

article.

1. Spontaneous Magnetostriction

Generally speaking, spontaneous magnetostriction is

deﬁned as that part of strain which is due to the onset

of a magnetic order in a material (du Tremolet de

Lacheisserie 1993).

A phenomenological approach consists of treating

the onset of spontaneous magnetization as a small

perturbation of a hypothetical nonmagnetic state. At

a given temperature, the perturbation is entirely de-

ﬁned by the magnetization direction cosines a

The spontaneous magnetostriction tensor S is then

expanded as a function of this perturbation. In a

convenient frame (Nye 1957), allowing for time re-

versal invariance, and using the usual Einstein con-

vention on repeated indices, this gives:

¼ Q

ijkl

þ even higher order termsði; j ¼ 1; 2; 3Þ

ð1Þ

The tensor Q and the higher-order tensors are ex-

pected to be temperature-dependent. Consideration

of the ﬁrst term only is sufﬁcient to account for the

properties of most materials, so that only the form of

Q will be considered in the following. Since Q is a

polar tensor of order 4, it follows the same symmetry

rules as the elastic tensors (Nye 1957). Two cases are

worth considering: cubic crystals and isotropic ma-

terials. In the following, emphasis will be placed on

cubic symmetry since it contains the isotropic situa-

tion as a special case.

In cubic crystals, the convenient reference frame is

obviously the one that coincides with crystallographic

axes.

There are only 15 nonzero components of Q out of

which only three are independent, namely

1111

¼Q

2222

¼ Q

3333

¼ Q

1122

¼Q

2211

¼ Q

1133

¼ Q

3311

¼ Q

2233

¼ Q

3322

¼ Q

2323

¼Q

3232

¼ Q

1313

¼ Q

3131

¼ Q

1212

¼ Q

2121

¼ Q

ð2Þ

so that the components of spontaneous magneto-

striction can now be written as:

¼ðQ

 Q

Þa

þ Q

;

¼ðQ

 Q

Þa

þ Q

;

¼ðQ

 Q

Þa

þ Q

¼ S

¼ Q

; S

¼ S

¼ Q

;

¼ S

¼ Q

ð3Þ

In isotropic material the reference frame can be cho-

sen arbitrarily. The relevant Q components and the

spontaneous magnetostriction are immediately de-

rived from Eqns. (2) and (3), respectively, by ex-

pressing the isotropy relation, Q

¼Q

Q

2. Volume Magnetostriction

The relative volume expansion of a fully saturated

sample with respect to the nonmagnetic reference

state is DV/V ¼S

þS

, which, from Eqn. (3),

turns out to be equal to Q

þ2Q

whatever the ori-

entation a

. It can be concluded that in cubic

crystals (and therefore in isotropic materials) volume

magnetostriction is actually independent of the tech-

nical magnetization state and equal to Q

þ2Q

3. Joule Magnetostriction

Joule magnetostriction is by definition the deforma-

tion of a magnetic material which is induced at a ﬁxed

temperature by those technical magnetization pro-

cesses that involve a rotation of the spontaneous

magnetization.

It is interesting from an experimental viewpoint

to redeﬁne the zero deformation reference state as

the isotropic demagnetized state instead of the

hypothetical nonmagnetic state. By this, it is meant

a multidomain state with an isotropic distribution of

837

Magnetostrictive Materials

the magnetization orientation and, of course, van-

ishing average (or technical) magnetization. The

strain components as measured with respect to this

new reference state (called for convenience sponta-

neous Joule deformations) are then:

¼ðQ

 Q

Þða

 1=3Þ;

¼ðQ

 Q

Þða

 l=3Þ;

¼ðQ

 Q

Þða

 1=3Þ

¼S

¼ Q

; S

¼ S

¼ Q

;

¼S

¼ Q

ð4Þ

Joule magnetostriction in a cubic crystal may thus

be characterized by only two parameters, namely

Q

) and Q

. They can be determined, at least

in principle, by measuring the relative elongation

l(a

) along various magnetization directions

when the sample is brought from the isotropic de-

magnetized state to the saturated state. Two partic-

ular orientations are worth considering, /100S and

/111S: a simple calculation gives

100

¼ 2=3ðQ

 Q

Þ; l

111

¼ 2=3Q

so that the expression of the spontaneous Joule de-

formations can be rewritten as:

¼3=2l

100

ða

 1=3Þ; S

¼ 3=2l

100

ða

 1=3Þ;

¼3=2l

100

ða

 1=3Þ; S

¼ 3l

111

;

¼3l

111

; S

¼ 3l

111

ð5Þ

Here we have used the contracted technical nota-

tion for strains (Nye 1957): S

¼S

þS

¼S

þS

¼S

þS

¼S

In practice, due to the difﬁculty of actual realization

of an isotropic demagnetized state, the preferred

method of measurement relies on the strain variations

caused by controlled modiﬁcations of the orientation

of fully saturated states (du Tremolet de Lacheisserie

1993).

In isotropic materials, Joule magnetostriction is

characterized by only one parameter since l

100

111

¼l

where l

is called the saturation magneto-

striction.

4. Free Energy Density and Other

Magnetostrictive Effects

The knowledge of equations of state is essential to

most applications of a material. This is related to the

knowledge of its free energy density function F. The

free energy density function of a single domain cubic

magnetostrictive crystal at a given temperature can

be easily calculated for a given elastic and saturated

magnetic state speciﬁed by stress components

¼T

, T

¼T

, T

¼T

, T

¼T

, T

¼T

and magnetization di-

rection cosines a

, a

. Calculation can be made in

two different ways. First the crystal is saturated in the

speciﬁed directions a

, a

allowing it to deform

freely (zero applied stress), so that it takes the spon-

taneous Joule deformations (Eqn. (5)). Then suitable

increasing stresses are applied to the sample in order

to bring it reversibly into its speciﬁed elastic state

while blocking the magnetization. The result is:

F ¼K

ða

; a

Þþ1=2s

½T

þ T



þ s

½T

þ T

þ1=2s

½T

þ T

ð6Þ

where K

, a

) is the zero stress anisotropy en-

ergy of the crystal (deﬁned to within a constant) and

the s

are the elastic compliances at blocked magnet-

ization.

Alternatively, the stress T canbeappliedinthe

demagnetized state, and then the magnetization is

brought to state a

which still produces the spon-

taneous deformation (Eqn. (5)). Therefore, the total

work supplied to the material is:

F ¼1=2s

½T

þ T

þs

½T

þ T



þ 1=2s

½T

þ T

þK

ða

; a

þ 3=2l

100

½T

ða

 1=3Þ

þ T

ða

 1=3ÞþT

ða

 1=3Þ

þ 3l

111

½T

þ T

ð7Þ

A comparison of Eqns. (6) and (7) shows that the an-

isotropy energy density under stress T is given by

ða

; a

Þ¼K

ða

; a

Þþ3=2l

100

½T

ða

 1=3Þ

þ T

ða

 1=3ÞþT

ða

 1=3Þ

 3l

111

½T

þ T

ð8Þ

Finally, noticing that T

¼c

(SjSj

)where

i,j ¼16, and putting 3/2 (c

c

100

¼B

and

3c

111

¼B

, it is easy to obtain from Eqns. (5) and

(6):

F ¼K

ða

; a

Þþ1=2C

½S

þ S



þ c

½S

þ S



þ 1=2c

½S

þ S

þB

½S

ða

 1=3Þ

þ S

ða

 1=3Þ

þ S

ða

 1=3Þ þ B

½S

þ S



ð9Þ

where the c

are the elastic stiffnesses, and K

, a

) is the anisotropy energy at zero strain which, for

838

Magnetostrictive Materials

symmetry reasons, has the same form as K

and B

are called magnetoelastic coupling coefﬁcients.

For an isotropic material, since l

100

¼l

111

¼l

and

c

¼2c

, Eqn. (9) reduces to

F ¼K

ða

; a

Þþ1=2c

½S

þ S



þ c

½S

þ S

þ1=2c

½S

þ S



þ B½S

ða

 1=3ÞþS

ða

 1=3ÞþS

ða

 1=3Þ

þ S

ð10Þ

where B ¼3l

is the unique magnetoelastic cou-

pling coefﬁcient.

4.1 Inverse Magnetostrictive Effect

Equation (8) shows that the application of a stress on

a magnetostrictive sample induces an additional an-

isotropy. This is the actual mechanism of the inverse

effect since in general this added anisotropy will pro-

duce a reorientation of the magnetization.

4.2 DE Effect

The effective elastic constants of a magnetostrictive

material are generally modiﬁed by the application of

a static magnetic ﬁeld (DE effect). The reason is that

the effective elastic constants are different depending

on whether the magnetization is free to rotate (at zero

d.c. ﬁeld) or bound to a given direction (at high d.c.

ﬁeld). Calculation of the DE effect is easy only in

simple domain conﬁgurations. A particularly simple

example is that of a single domain cubic crystal bear-

ing an anisotropy of the usual form K

) ¼

K

þa

] with K

40, magnetized

along 100 by a static ﬁeld H. Starting from Eqn.

(9), the differential shear stiffness can be given by

=@S

¼ @T

=@S

¼ c

 B

=ðK

þ 1=2 m

HÞ

As expected, it tends to the blocked magnetization

value at high ﬁeld and decreases when the applied

ﬁeld is reduced. A measure of the DE effect is the

relative variation controlled by the applied ﬁeld, i.e.,

/(K

) ¼k

(Nye 1957). The interest in this pa-

rameter and the reason for this notation will be made

clear in the next section.

4.3 Nonlinear elasticity

In correlation with the DE effect, strong nonlinear

elastic effects are also observed in high coupling factor

materials. The source of this elastic nonlinearity lies in

the nonlinearity of the magnetic response itself. Some

calculations can be made for simple single-domain

conﬁgurations but they will not be developed here.

5. Magnetostriction of Unsaturated States:

Piezomagnetic Formulation

Below saturation, the magnetostrictive response of

polycrystalline and amorphous materials involves do-

main processes which can be quite complicated. The

usual method, at least for application purposes, is to

characterize the material properties only around the

relevant bias point as far as possible by phenomeno-

logical linear and reversible equations of states.

In general, both a bias ﬁeld and a bias uniaxial

stress are applied to the material along the same di-

rection. The material has then uniaxial symmetry

around the bias direction which is taken as axis three

of a rectangular frame. In this frame the linearized

properties are therefore described by matrix relations

that are analogous to those of piezoelectric ceramics

(Nye 1957). Taking, for example, differential ﬁeld h,

and differential stress T as independent variables, the

equations of state are written as

S ¼ sT þ d  h; b ¼ dT þ gh ð12Þ

Here, taking advantage of the contracted notation,

T and the differential strain S are six-component

vectors whereas b, the differential induction, and h

are three-component vectors. d is thus a three line, six

columns rectangular matrix. d* is actually the trans-

position of d, a consequence of the reversibility

assumption.

The form of the piezomagnetic matrix d is the same

as its piezoelectric analogue, i.e., there are only ﬁve

nonzero components out of which only three are in-

dependent, namely d

¼d

, and d

In general in transducer conﬁgurations, stress and

strain are collinear single-component vectors, as are

the ﬁeld and induction. Vector Eqn. (12) then reduces

to a system of two scalar linear equations (no sum-

mation on repeated subscripts!):

¼ s

þ d

; b

¼ d

þ m

It is convenient to deﬁne a dimensionless coefﬁcient

¼(d

)

1/2

called the magnetomechanical cou-

pling factor. The reason is that its square actually

measures the ratio of output to input energy in a

static reversible conversion process. By a conversion

process, it can deﬁne a transformation of magnetic

energy into mechanical energy or vice versa. Three

useful magnetomechanical coupling factors may be

deﬁned in uniaxial magnetostrictors, namely:

¼ðd

1=2

;

¼k

¼ðd

1=2

; and

¼k

¼ðd

1=2

839

Magnetostrictive Materials