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

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natural antiferromagnet is that in this way it is easier

to prepare an uncompensated surface with a net

magnetic moment. This is of interest in the context of

‘‘exchange anisotropy’’ used for pinning the magnet-

ization unidirectionally. At the surface of a natural

antiferromagnet the moment can be largely compen-

sated, due to surface roughness or other irregularities.

Artiﬁcial antiferromagnets can replace natural ones,

for example, in ﬁeld sensors, where they are used to

shift remagnetization curves using the exchange an-

isotropy effect. The application is displayed in the

case of a rotational GMR type sensor in Fig. 7.

See also: Magneto-optic Multilayers

Bibliography

Bruno P 1995 Theory of interlayer magnetic coupling. Phys.

Rev. B 52, 411–39

rgler D, Demokritov S, Gru

nberg P, Johnson M 2001 In-

terlayer exchange coupling in transition metal systems. In:

Buschow K H J (ed.) Magnetic Materials: Vol. 13. Elsevier,

Amsterdam

Demokritov S O 1998 Biquadratic interlayer coupling in lay-

ered magnetic systems. J. Phys. D: Appl. Phys. 31, 925–41

Gru

nberg P, Schreiber R, Pang Y, Brodsky M B, Sowers H

1986 Layered magnetic structures: evidence for antiferro-

magnetic coupling of Fe layers across Cr interlayers. Phys.

Rev. Lett. 57, 2442–5

Majkrzak C F, Cable J W, Kwo J, Hong M, Mc Whan D B,

Yafet Y, Waszczak J V, Vettier C 1986 Observation of a

magnetic antiphase domain structure with long-range order

in a synthetic Gd-Y superlattice. Phys. Rev. Lett. 56, 2700–3

Majkrzak C F, Kwo J, Hong M, Yafet Y, Doon Gibbs, Chien

C L, Bohr J 1991 Magnetic rare earth superlattices. Adv Phys.

40, 99–189

Parkin S S P, More N, Roche K P 1990 Oscillations in exchange

coupling and magnetoresistance in metallic superlattice struc-

tures: Co/Ru, Co/Cr and Fe/Cr. Phys. Rev. Lett. 64, 2304–7

Pierce D T, Unguris J, Celotta R J, Stiles M D 1999 Effect of

roughness, frustration, and antiferromagnetic order on mag-

netic coupling of Fe/Cr multilayers. J. Magn. Magn. Mater.

200, 290–321

Rhyne J J, Erwin R W, Borchers J, Salamon M B, Du R, Flynn

C P 1989 Neutron scattering studies of rare earth magnetic

multilayers. Physica B 159, 111–28

hrig M, Scha

fer R, Hubert A, Mosler R, Wolf J A, Demo-

kritov S, Gru

nberg P 1991 Domain observations in Fe-Cr-Fe

layered structures: evidence for a biquadratic coupling effect.

Phys. Stat. Solidi A 125, 635–56

Salamon M B, Sinha S, Rhyne J J, Cunningham J E, Erwin R

W, Borchers J, Flynn C P 1986 Long range incommensurate

magnetic order in a Dy–Y multilayer. Phys. Rev. Lett. 56,

259–62

Slonczewski J C 1995 Overview of interlayer exchange theory.

J. Magn. Magn. Mater. 150, 13–24

Stiles M D 1999 Interlayer exchange coupling. J. Magn. Magn.

Mater. 200, 322–37

Unguris J, Celotta R J, Pierce D T 1991 Observation of

two different oscillation periods in the exchange coupling of

Fe/Cr/Fe(100). Phys. Rev. Lett. 67, 140–3

van den Berg H A M, Clemens W, Gieres G, Rupp G, Schelter

W, Vieth M 1996 GMR sensor scheme with artiﬁcial antif-

erromagnetic subsystem. IEEE Trans. Magn. 32 (5), 4624–6

P. A. Gru

nberg

Forschungszentrun Ju

lich, Germany

D. T. Pierce

National Institute of Standards and Technology

Gaithersburg, Maryland, USA

Muon Spin Rotation (lSR): Applications

in Magnetism

Following the pioneer work by Gurevich et al.in

1972, demonstrating the possibility to use polarized

positive muons in the ﬁeld of metal physics, numer-

ous experiments using the Muon Spin Rotation/

Relaxation (mSR) technique have been successfully

undertaken with special emphasis on the study of

magnetic phenomena in matter. Although some ex-

perimental and theoretical problems remain to be

solved, it is fair to say that the mSR technique became

widely recognized as a mature technique able to fur-

nish a complementary point of view when investiga-

ting magnetism in matter.

The interest on this novel technique is based on the

complementarity between the particular information

extracted from the mSR studies and the one obtained

from either more classical microscopic techniques

(e.g., neutron diffraction, nuclear magnetic resonance

(NMR), and Mo

ssbauer spectroscopy) or bulk stud-

ies such as specific heat or magnetic susceptibility

measurements. The details of the technique itself

are extensively discussed in numerous review articles

(e.g., Schenck 1985) and correspondingly, several

articles provide extensive reviews of the magnetic

materials investigated by mSR spectroscopy (e.g.,

Schenck and Gygax 1995, Dalmas de Re

otier and

Yaouanc 1997). A concise description of the method

can be also found in Cox and Stoneham 1993.

1. The lSR Signal—The Basics

A polarized m

beam is obtained by collecting the

muons produced via the two-body decay of positive

pions. Due to the zero value of the pion spin, the

muon beam is 100% polarized (helicity 1) in the

pion frame. After the implantation into the sample,

the muons are decelerated within 100 ps and come to

rest at an interstitial site without any significant loss

of polarization. Subsequently, if the m

are subject to

magnetic interactions, their polarization becomes

time dependent (P

(t)). Due to the weak decay of

the muon (average lifetime t

B2.2 ms), the time

960

Muon Spin Rotation (lSR): Applica tions in Magnetism

evolution of P

(t) can be deduced by measuring the

distribution as a function of time of the positron

emitted after the muon decay (the positron is emitted

preferentially along the direction of the decaying

muon spin). In the time differential mSR technique,

repeated measurements (B1 10

) are made of the

time interval between the m

implantation into the

sample and the detection in a particular direction of

the emitted positron.

The time histogram of the collected intervals has

the form

eþ

ðtÞ¼B þ N



1 þ A

ðtÞ

ð0Þ



ð1Þ

where the last part of the second term represents

the asymmetric positron emission (with typically AB

0.3). B is a time-independent background, N

is a

normalization constant, and the exponential ac-

counts for the decay of the m

. P

(t) is deﬁned

as the projection of P

(t) along the direction of the

initial polarization, i.e., P

(t) ¼P

(t)  P

(0)/P

(0) ¼

G(t)P

(0), where G(t) reﬂects the normalized m

spin

auto-correlation function

GðtÞ¼

/SðtÞSð0ÞS

Sð0Þ

ð2Þ

which depends on the average value, distribution, and

time evolution of the internal ﬁelds and, therefore,

contains all the physics of the magnetic interactions

of the m

inside the sample.

Equation (1) can therefore be written as:

eþ

ðtÞ¼B þ N



1 þ AGðtÞ½ð3Þ

where AG(t) is often called the mSR signal and the

envelop of G(t) is known as the m

depolarization

function. In the case of static local magnetic ﬁelds

) at the m

site, G( t) is given by:

GðtÞ¼

f ðB

Þ½cos

y þ sin

y cosðg

tÞdB

ð4Þ

where f(B

) is the magnetic ﬁeld distribution function,

y is the angle between the local ﬁeld and P

(0), B

7 and g

/(2p) ¼13.553879 (70.7 ppm) kHzG

1

the gyromagnetic ratio of the muon. Therefore, the

careful analysis of the mSR signal will provide infor-

mation on the local ﬁeld distribution at the interstitial

site where the muon came at rest.

1.1 Transverse-ﬁeld (TF) Technique—Muon Spin

Rotation

For this technique, an external ﬁeld H

ext

is applied

perpendicular to the initial muon polarization P

(0).

(t) processes around the total ﬁeld B

at the muon

site, the value of which can be extracted from the

oscillatory component of G(t), recalling that the m

frequency n

¼o/(2p) ¼g

/(2p). In addition to the

determination of the muon Knight shift, this tech-

nique allows one to gain more insight on the inho-

mogeneous ﬁeld distribution at the muon site caused

by static ﬁelds and on the presence of 1/T

processes

responsible for homogeneous line broadening.

1.2 Zero-ﬁeld (ZF) and Longitudinal-ﬁeld (LF)

Techniques—Muon Spin Relaxation

With the ZF technique one monitors the time evolu-

tion of the muon ensemble under the action of inter-

nal magnetic ﬁelds. This technique is widely utilized to

measure the spontaneous m

Larmor frequencies in

magnetically ordered phases, providing information

about the values of the static moment and the type of

magnetic structures. The observation of spontaneous

frequencies is, however, limited to systems where

the presence of static electronic moments produces

well-deﬁned local ﬁelds at the m

stopping site.

Hence, the m

spin autocorrelation function G(t)

depends sensitively on details of the magnetic struc-

ture and of course on the m

stopping site.

In a ZF experiment, a depolarization can occur due

to the presence either of a static distribution of inter-

nal ﬁelds arising from static nuclear or electronic di-

pole ﬁelds, or of a time dependence of the internal

ﬁelds at the muon site. This time dependence may

arise from a ﬂuctuation of the magnetic moments or

from muon diffusion. In contrast, in a LF conﬁgura-

tion (i.e., H

ext

(0)), by choosing H

ext

to be stronger

than the internal ﬁelds, the muon’s states ‘‘up’’ and

‘‘down’’ are eigenstates of the Zeeman Hamiltonian

and any inhomogeneous (static) distribution of the

internal ﬁelds will not affect the time evolution of the

polarization, which will remain constant. This

behavior reﬂects the decoupling of the m

spin from

the static internal ﬁelds. Hence, to pinpoint the origin

of the depolarization, decoupling experiments in a

longitudinal ﬁelds are routinely performed.

2. Why Muons?

There are several advantages in using the mSR tech-

nique to study magnetic systems.

(i) Due to the large muon magnetic moment

¼8.89 m

), the technique is sensitive to extremely

small internal ﬁelds (down to B1 10

5

T) and,

therefore, can probe local magnetic ﬁelds which can

be nuclear or electronic in nature.

(ii) The local probe character of the muon makes

mSR very sensitive to spatially inhomogeneous mag-

netic properties. Hence the occurrence of different

phases in the sample will be reﬂected by different

961

Muon Spin Rotation (lSR): Applications in Magnetism

components in the mSR signal, i.e.,

AGðtÞ¼

ðtÞð5Þ

(iii) A careful analysis of the amplitude A

of these

components furnishes a direct measure of the fraction

of the sample involved in a particular phase (with the

condition S

¼A). Similarly, mSR is a powerful tool

when magnetic order is of short range and/or random

nature where neutron experiments will fail.

(iv) In addition, the mSR technique can be utilized

to check the coexistence of different types of ground

states at the microscopic level.

(v) In magnetic phases, the possibility given by mSR

to independently extract the value of the ordered

static moment and the sample volume involved in the

magnetic phase allows one to overcome a shortcom-

ing of other techniques (such as the neutron scattering

technique) for which these two parameters are strong-

ly coupled, therefore, hampering their absolute deter-

mination.

(vi) Since no applied ﬁeld is necessary to polarize

the spin of the implanted muons, mSR measurements

can be made in the absence of a perturbative external

ﬁeld.

(vii) The muon is a spin-

particle and consequently

is free of quadrupolar interactions. This greatly sim-

pliﬁes the analysis of mSR, with a reduced set of ad-

justable parameters.

In the following, some exemplary studies illustrat-

ing some of the above described advantages of the

mSR technique will be briefly discussed.

2.1 Small Values of the Ordered Magnetic Moments

The capability of mSR to detect weak static magnetism

has been illustrated by the discovery of magnetic

groundstatesinso-calledheavy-fermion systems

that were long considered examples of Fermi-liquid

paramagnets. In these systems, composed by local

magnetic moments, the competition between the de-

magnetizing on-site Kondo interaction and the inter-

site Ruderman–Kittel–Kasuya–Yosida interaction

leads to a richness of different ground states (see al-

so Heavy-fermion Systems; Kondo Systems and Heavy

Fermions: Transport Phenomena). An example is the

occurrence of itinerant magnetism of the heavy quasi-

particles, which is characterized by extremely small

static magnetic moments (of the order of 1 10

3

Figure 1 exhibits the depolarization rate of mSR

spectra obtained in zero-applied ﬁeld for the heavy-

fermion compound CeRu

(Amato et al. 1994).

Below 2 K, the depolarization rate exhibits a signif-

icant increase corresponding to an enhancement

of the ﬁeld spread at the m

site of the order of

2 10

5

T, which originates from the electronic

moments. Considering that the different magnetic

structures observed in isostructural compounds with

light rare earth metals lead to nonzero net dipole ﬁeld

at the m

site, it was concluded that the ﬁeld distri-

bution with zero average value detected by mSR in

CeRu

was due to static Ce moments of the order

of m

B1 10

3

/Ce ordering in a complicated (pos-

sibly incommensurate) structure.

2.2 Coexisting Phases

In different classes of systems, the question of the

coexistence of magnetism and superconductivity has

attracted considerable attention. This is, for example,

the case for the heavy-fermion compounds where

both ground states are found to microscopically co-

exist for U-based systems and to occur in different

domains in Ce-based systems. Such a conclusion was

mainly extracted from mSR studies by carefully anal-

yzing the potential presence of different components

in the spectra which could reﬂect the coexistence in a

sample of domains with different magnetic properties

(muons stopping in different domains will contribute

to different components of the signal.)

An interesting example of such studies is provided

by the measurements on the heavy-fermion compound

CeCu

(Feyerherm et al. 1997) which, at

the opposite of what was concluded from thermody-

namic studies only, gave strong evidence against a

microscopic coexistence of superconductivity and

magnetism. The conclusions drawn from the zero-

ﬁeld mSR data are based on the observation of two

components in the m

depolarization function (see

Fig. 2). One of the components shows a fast relaxation

and is associated with magnetic domains developing

below B1.3 K. The other exhibits a slow relaxation

Figure 1

Temperature dependence of the ZF (open symbols) and

LF (B

ext

¼0.6 T, closed symbols) depolarization rates in

CeRu

measured in two monocrystalline samples

(A and B). The initial m

-polarization was along the

tetragonal c-axis.

962

Muon Spin Rotation (lSR): Applica tions in Magnetism

and reﬂects paramagnetic domains which become su-

perconducting below B0.6 K. Systematic mSR studies

have shown that the increase of the magnetic fraction

(i.e., the amplitude A

of the corresponding compo-

nent of the mSR signal) is strongly sample dependent

and that the amount of magnetic phase is closely re-

lated to the sample-dependent size of the specific heat

jump at the superconducting transition.

2.3 Short-range Magnetism

In the family of the high T

(and magnetically related)

compounds, some significant information was ob-

tained from the mSR technique. In addition to the

ﬁrst indication of magnetic order in several com-

pounds, the mSR technique has played a key role in

the elucidation of phase diagrams due to its capabil-

ity to detect short-range and random magnetism. An

example is the studies on La

CuO

4d

, for which the

appearance of magnetism depends on the exact

oxygen content (the input of O

2

removes electrons

and creates holes in the system). Whereas some

oxygen deﬁciency stabilizes the magnetic state, an

excess of oxygen suppresses the antiferromagnetic

order and eventually leads to superconductivity.

Figure 3 shows the spontaneous muon spin preces-

sion frequency (n

) observed in zero ﬁeld for different

samples of La

CuO

4d

(Uemura et al. 1988). Since n

is proportional to the value of the static copper mo-

ment (m

), the data indicate that m

changes only

within B15% for the different samples despite the

large difference in the ordering temperatures

(from 15 K to 300 K). This is in contrast to results

from neutron scattering on three different monocrys-

talline samples (see Fig. 4) from which a sublattice

magnetization was derived from the intensity of the

magnetic Bragg peaks, assuming that the whole vol-

ume exhibits long-range magnetism.

The difference between mSR and neutron results

can be understood by considering the muon as a

point-like probe in real space. Therefore, the ﬁeld at

the muon site will be mostly due to its neighboring

copper moments, even if the spatial correlation char-

acterizing the magnetic order may be short range. In

contrast, the elastic magnetic scattering intensity

measured in a neutron scattering experiment will

probe the magnetic ordering in reciprocal space and

will, therefore, only appear when the spatial 3D or-

dering is long range. It can, therefore, be concluded

that the difference between mSR and neutron results

is mainly due to a loss of long-range magnetism with

increasing oxygen content (i.e., with increasing hole

Figure 3

Temperature dependence of the spontaneous muon-spin

precession frequency in La

CuO

2y

for various y (from

Uemura et al. 1988).

Figure 2

Time evolution of the m

polarization measured in zero-

ﬁeld in a CeCu

2.05

sample at low temperature. Note

the clear two-component structure of the signal

measured below 1 K.

Figure 4

Sublattice magnetization derived from the magnetic

Bragg-peak intensity of neutron scattering experiments

on La

CuO

2d

samples with various d (see Uemura et al.

1988 for details).

963

Muon Spin Rotation (lSR): Applications in Magnetism

concentration) introducing frustrations in the mag-

netic CuO

planes. Consequently, the value of the

static copper moment extracted from the neutron

data for samples with low T

is systematically un-

derestimated due to the reduced fraction of the spec-

imen exhibiting long-range magnetic order.

3. Intrinsic Difﬁculties of the Technique

The analysis of the mSR results can be hampered by

several potential difﬁculties.

To extract quantitative information, the knowl-

edge of the muon stopping site is usually mandatory.

To determine it unambiguously, precise Knight-shift

measurements are necessary along the principal cry-

stallographic directions. Therefore, and in addition to

the intrinsic difﬁculties of such measurements, the

muon site determination requires single crystals,

which might be difﬁcult to obtain for given com-

pounds or alloys.

Another potential problem is the presence of m

diffusion. Fortunately, observations show that diffu-

sion is usually restricted to high temperatures. Since

most aspects of magnetic phenomena occur at low

temperature, m

diffusion will usually be of limited

concern.

An intrinsic problem of mSR is the addition of

an extra positive charge in the lattice. This leads to a

modiﬁed local charge distribution, due to the screen-

ing of the muon’s positive charge, and to a slight

lattice relaxation, reﬂected by a small shift in the po-

sition of the nearest neighbor ions. This latter effect

can be exactly quantiﬁed by a careful comparison

between the data and simple theoretical calculations.

Another, more disturbing, effect of the implan-

tation of a positive charge in the lattice is that the

modiﬁcation of the local charge density can possibly

affect the crystal electric ﬁeld (CEF) of the neighbor-

ing atoms (see Localized 4f and 5f Moments: Mag-

netism). In systems based on rare-earth or actinides

elements (for which the splitting of the ground state

due to CEF is rather small) this effect could lead to a

slight change of the atomic magnetic susceptibility for

the neighboring ions. It has been shown that this ef-

fect can be important for singlet ground-state systems

where the splitting between the excited states and the

singlet ground state is very low (i.e., o1 meV). It is,

therefore, of primary importance to compare system-

atically, in each particular case, the atomic suscepti-

bility sensed by the interstitial muon to the bulk

susceptibility, to detect the potential presence of such

muon-induced effects.

4. Future Developments

4.1 Low Energy Muons

Thin ﬁlms, nanomaterials, and multilayered com-

pounds have emerged as significant elements in science

and technology. In response to the need of novel

techniques to investigate the magnetic properties of

these objects, the development of muon beams with

energy in the range of one eV to several keV was

needed. In this direction, promising results have been

obtained by slowing down energetic muons in appro-

priate moderators consisting of a thin layer of a van

der Waals gas frozen on a substrate (Morenzoni

1997). The availability of epithermal polarized muons,

therefore, opens the possibility to extend the mSR

technique to the study of thin ﬁlms and surfaces.

4.2 Muon on Request

Another promising evolution of the technique has

been the development of a muon on request setup,

combining, for the case of time-differential mSR ex-

periments, the advantages of continuous and pulsed

muon beams (Abela et al. 1999). Although continu-

ous muon beams possess a good time resolution,

limited mainly by the design of the detectors, the

presence of random background in the positron his-

tograms limits the useful time interval to about 10 ms,

excluding investigations of slow relaxation processes.

On contrast, pulsed muon beams deliver many muons

per pulse at low repetition rate (51/t

) essentially

without background on the histograms. However, the

time resolution for such beams is strongly limited by

the ﬁnite length of the muon pulses.

A method to solve the background problem at

continuous beam is to extract one muon at a time out

of a continuous beam by means of a fast-switching

electrostatic deﬂector, on request from a mSR instru-

ment. Since the positron histograms obtained are

essentially background free, this system opens, due

to the high time resolution, the unique possibility of

studying slow relaxation phenomena in high mag-

netic ﬁelds.

See also: Electron Systems: Strong Correlations;

Heavy-fermion Systems; Localized 4f and 5f

Moments: Magnetism; Magnetism in Solids: General

Introduction

Bibliography

Abela R, Amato A, Baines C, Donath X, Erne R, George D C,

Herlach D, Irminer G, Reid I D, Renker D, Solt G, Suhi D,

Werner M, Zimmermann U 1999 Muons on request

(MORE): combining advantages of continuous and pulsed

beam. Hyperﬁne Inter. 121, 575–8

Amato A, Feyerherm R, Gygax F N, Schenck A, Flouquet J,

Lejay P 1994 Ultrasmall-moment static magnetism in

CeRu

. Phys. Rev. B. 50, 619–22

Cox S F J, Stoneham A M 1993 Muon beams used for studying

the solid state. In: Cahn R W, Lifshin E (eds.) Concise En-

cyclopedia of Material Characterization. Pergamon, Oxford

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Muon Spin Rotation (lSR): Applica tions in Magnetism

Dalmas de Re

otier P, Yaouanc A 1997 Muon spin rotation and

relaxation in magnetic materials. J. Phys. Condens. Matter 9,

9113–66

Feyerherm R, Amato A, Geibel C, Gygax F N, Hellmann P,

Heffner R H, MacLaughlin D E, Mu

ller Reisener R,

Nieuwenhuys G J, Schenck A, Steglich F 1997 Competition

between magnetism and superconductivity in CeCu

Phys. Rev. B 56, 699–710

Gurevich I I, Meleshko E A, Muratova I A, Nikolsky B A,

Roganov V S, Selivanov V I, Sokolov B V 1972 Dipole in-

teractions and diffusion of m

meson in copper. Phys. Lett. A

40, 143–4

Morenzoni E 1997 Low energy muons as probes of thin ﬁlms

and surfaces. Appl. Magn. Res. 13, 215–29

Schenck A 1985 Muon Spin Rotation Spectroscopy. Hilger,

Bristol, UK

Schenck A, Gygax F N 1995 Magnetic materials studied by

muon spin spectroscopy. In: Buschow K H J (ed.) Handbook

of Magnetic Materials. Elsevier, Amsterdam, Vol. 9, pp.

57–302

Uemura Y J, Kossler W J, Kempton J R, Yu X H, Schone H E,

Opie D, Stronach C E, Brewer J H, Kieﬂ R F, Kreitzman S

R, Luke G M, Riseman T, Williams D L, Ansaldo E J,

Endoh Y, Kudo E, Yamada K, Johnston D C, Alvarez M,

Goshorn D P, Hidaka Y, Oda M, Enomoto Y, Suzuki M,

Murakami T 1988 Comparison between muon spin rotation

and neutron scattering studies on the 3-dimensional magnetic

ordering of La

CuO

4d

. Physica C 153–5, 769–70

A. Amato

Paul Scherrer Institute, Villigen, Switzerland

965

Muon Spin Rotation (lSR): Applications in Magnetism

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966

Nanocrystalline Materials: Magnetism

It is well known that the microstructure, noticeably

the grain size, essentially determines the hysteresis

loop of a ferromagnetic material. Figure 1 summa-

rizes the typical variation of the coercivity, H

, over

the whole range of structural correlation lengths

starting from atomic distances in amorphous alloys

over grain sizes, D, in the nanometer region up to

macroscopic grain sizes. The permeability shows an

analogous behavior being essentially inversely pro-

portional to H

. The 1/D-dependence of coercivity for

large grain sizes (Pfeifer and Radeloff 1980) reﬂects

the conventional rule that good soft magnetic prop-

erties require very large grains (D4100 mm). Thus,

the reduction of particle size to the region of the do-

main wall width increases H

towards a maximum

controlled by the anisotropies present. Accordingly,

ﬁne-particle systems have been mostly discussed as

hard magnetic materials (Luborsky 1961). The lowest

coercivities, however, are again found for the smallest

structural correlation lengths, as in amorphous alloys

(‘‘grain size’’ of the order of atomic distances) and in

nanocrystalline alloys for grain sizes Do20nm. The

extraordinary D

-dependence of coercivity at small

grain sizes, moreover, demonstrates how close soft

and hard magnetic behavior actually can be. Indeed,

the soft magnetic behavior on which we will focus in

this article is only one manifestation of the novel and

extraordinary magnetic properties that can be real-

ized by establishing structural features on the nano-

meter scale. Thus, nanocrystalline microstructures

are also of great interest because of their ability to

enhance the properties of rare earth hard magnets

(Buschow 1997).

The decrease of coercivity shown in Fig. 1 for small

grain sizes has to be distinguished from superpara-

magnetic phenomena, i.e., the well-known decrease of

coercivity in small, isolated or weakly coupled par-

ticles owing to thermal excitation (Kneller 1969,

Luborsky 1961). In the present case we deal with

small ferromagnetic crystallites well coupled by ex-

change interaction which results in low coercivity

and, unlike superparamagnetic particles, in a simul-

taneously high permeability.

The most prominent examples of soft magnetic

nanocrystalline materials are devitriﬁed glassy Fe–

Cu–Nb–Si–B alloys (Yoshizawa et al. 1988); further

examples are devitriﬁed Fe–(Cu)–Zr–B alloys (Suzuki

et al. 1991) or Fe–Hf–C thin ﬁlms (Hasegawa and

Saito 1991). The alloy composition originally pro-

posed and subsequently not much changed is

73.5

13.5

and can be considered as a typ-

ical FeSiB metallic glass composition with additions

of copper and niobium. The material is produced by

rapid solidiﬁcation of an originally amorphous rib-

bon, typically 20 mm thick, which is subsequently an-

nealed above its crystallization temperature. The

resulting microstructure is characterized by random-

ly oriented, ultraﬁne grains of b.c.c. Fe–20at.% Si

with typical grain sizes of 10–15nm, hereafter referred

to as nanocrystalline. The grains are embedded in

a residual amorphous matrix which occupies about

20–30% of the volume and separates the crystallites

at a distance of about 1–2 nm. The formation of this

particular structure is ascribed to the combined ef-

fects of copper and niobium (or other group 4 to 6

elements) and their low solubility in b.c.c. iron. Cop-

per promotes the nucleation of the b.c.c. grains, while

niobium hinders their growth and at the same time

inhibits the formation of boride compounds. The

structure and the associated magnetic properties are

astonishingly insensitive to the precise annealing con-

ditions over a wide range of annealing temperatures

between about 500 1C and 600 1C. Only annealing

at higher temperatures leads ﬁrst to the precipitation

of borides and subsequently to grain coarsening, both

of which deteriorate the soft magnetic properties

(Herzer 1997).

This article reviews the key mechanisms relevant to

the soft magnetic properties in the nanocrystalline

region. The materials highlighted will be the Fe–Cu–

Nb–Si–B alloys which are widely used. They are of

particular interest because of their superior soft mag-

netic properties compared to those of permalloys and

cobalt-based amorphous alloys. The general features,

however, also apply to other nanocrystalline mag-

netic materials.

Figure 1

Coercivity, H

, vs. grain size, D, for various soft

magnetic metallic alloys: Fe–Nb–Si–B (m), Fe–Cu–Nb–

Si–B (), Fe–Cu–V–Si–B (X), Fe–Zr–B (&), Fe–Co–Zr

(B), NiFe alloys (1, n), and Fe–6.5wt.% Si (J).

967

1. Magnetocrystalline Anisotropy and

Exchange Energy

The major requirement for superior soft magnetic be-

havior generally is a low or vanishing magnetocrys-

talline anisotropy. For example, in b.c.c. Fe–20at.%

Si, the constituent phase in nanocrystalline Fe

73.5

13.5

, the anisotropy constant is K

8kJm

3

, i.e., far too large to explain the low co-

ercivity (H

o1Am

1

) and high permeability (m

E10

)

by itself.

The key to understanding is that the effective an-

isotropy contribution of the small, randomly oriented

b.c.c. grains is essentially reduced by exchange inter-

action. The critical scale where the exchange energy

starts to balance the anisotropy energy is given by:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

A=K

ð1Þ

where A is the exchange stiffness constant, which is

about L

E 35nm for the material parameters of b.c.c.

Fe–20at.% Si. L

represents the minimum length scale

over which the direction of the magnetic moments can

vary appreciably and, for example, determines the or-

der of the domain wall width. Thus, if the grain size,

D, is reduced below L

, the magnetization does not

follow the randomly oriented easy axis of the individ-

ual grains, but increasingly is forced to align parallel

by exchange interaction. As a consequence, the effec-

tive anisotropy is an average over several grains and,

thus, considerably reduced in magnitude.

1.1 The Random Anisotropy Model

The degree to which the magnetocrystalline anisot-

ropy is ﬁnally averaged out has been successfully ad-

dressed in terms of the so-called random anisotropy

model (Herzer 1990), which was originally developed

in order to explain the soft magnetic properties of

amorphous ferromagnets by Alben et al. (1978). The

basic idea starts from an assembly of exchange cou-

pled grains of size D and volume fraction v

, with

magnetocrystalline anisotropies oriented at random

and embedded in an ideally soft ferromagnetic ma-

trix. The effective anisotropy constant, /KS, rele-

vant to the magnetization process results from

averaging over the N ¼v

/D)

grains within the

ferromagnetic correlation volume, V ¼ L

, deter-

mined by the exchange length, L

, i.e., the scale on

which exchange interaction forces the magnetic mo-

ments to align parallel. For a ﬁnite number, N,of

grains within the exchange volume, there will be al-

ways some easiest direction determined by statistical

ﬂuctuations. Thus, the averaged anisotropy energy

density, /KS, is determined by the mean ﬂuctuation

amplitude of the anisotropy energy of the N grains:

/KSE

ﬃﬃﬃﬃ

ﬃﬃﬃﬃﬃﬃ

ðD=L

3=2

ð2Þ

As the local magnetocrystalline anisotropies are

averaged out this way, the scale on which exchange

interaction dominates expands at the same time. Thus,

the exchange length, L

, has to be renormalized

by substituting /KS for K

in Eqn. (1); i.e., L

self-consistently related to the averaged anisotropy by:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

A=/KS

ð3Þ

The combination of Eqns. (2) and (3) then yields for

grain sizes smaller than the exchange length:

/KSEv

ðD=L

¼ v

ð4Þ

The most significant feature predicted by the ran-

dom anisotropy model is the strong variation of the

effective anisotropy, /KS, with the sixth power of

the grain size. Thus, for DEL

/3, i.e., grain sizes of

the order of 10–15 nm, the magnetocrystalline anisot-

ropy is reduced by three orders of magnitude towards

a few joules per cubic meter, i.e., small enough to en-

able superior soft magnetic behavior. Corresponding-

ly, the renormalized exchange length, L

,expands

into the micrometer region and, thus, is almost two

orders of magnitude larger than the natural exchange

length L

. High-resolution Kerr effect studies of nano-

crystalline Fe

73.5

13.5

indeed reveal very

wide domain widths of approximately 2 mm(Scha

fer

et al. 1991).

If there are no other anisotropies, coercivity and

initial permeability are closely related to /KS by:

¼ p

/KS

; m

¼ p

/KS

ð5Þ

where J

is the average saturation magnetization of

the material, and p

and p

are dimensionless prefac-

tors close to unity. These relations, obvious for co-

herent magnetization rotation, in the region D5L

also apply for domain wall displacements—actually,

on the scale of the 10nm small grains the magneti-

zation vector appears to rotate coherently if a 2 mm

(EL

) wide domain wall passes by.

The predictions of the random anisotropy model

provide a good guiding principle through the mag-

netic properties for grain sizes below about L

40–50nm. Thus, the D

-dependence is well reﬂected in

the coercivity (Fig. 1) and the initial permeability

(Herzer 1997).

1.2 Remanence Enhancement

The hysteresis loops for DoL

typically show an en-

hanced remanence (J

40.90), which is another

very characteristic feature for the dominance of ex-

change interaction in this small grain size region.

Actually, this remanence enhancement by exchange

interaction is most attractive for novel, isotropic,

nanoscale hard magnets (Buschow 1997). Yet at the

smallest grain sizes, when the random anisotropy is

968

Nanocrystalline Materials: Magnetism

sufﬁciently washed out, more long-range uniaxial

anisotropies take over and the remanence ratio de-

creases again to values around 0.5. Relevant to this

are magnetoelastic and ﬁeld-induced anisotropies,

which limit the beneﬁt of further grain size reduction

(dashed horizontal line in Fig. 1 for Do10nm).

For further details of remanence enhancement in

hard magnetic materials see Magnets: Remanence-

enhanced.

1.3 Grain Coupling

The crucial role of the exchange interaction for the

magnetic softening at DoL

also becomes most ev-

ident from the temperature dependence of the mag-

netic properties, shown in Fig. 2. Thus, when the

temperature approaches the Curie temperature, T

,of

the intergranular amorphous phase (T

B200–

300 1C), which is much lower than the T

of the

b.c.c. grains (T

B600 1C), the exchange coupling be-

tween the crystallites is largely reduced. As a conse-

quence, the initial permeability then decreases by

almost two orders of magnitude and the coercivity

increases correspondingly. Simultaneously the do-

main structure changes from wide domains to a pat-

tern of small, irregular domains (Scha

fer et al. 1991).

With increasing measuring temperature the co-

ercivity reaches a maximum and, ﬁnally, decreases

towards zero. Interestingly, the latter occurs at a

temperature (560 1C) below T

(607 1C) of the b.c.c.

grains. This indicates the transition to superpara-

magnetic behavior, which has been conﬁrmed by a

more detailed analysis of Lachowicz and Slawska-

Waniewska (1994).

Thus, we ﬁnd within a single sample a variety of

phenomena ranging from soft, to hard, to ﬁnally su-

perparamagnetic behavior being controlled by the

strength of the interaction between the particles.

2. Magnetostriction

Given a vanishing magnetocrystalline anisotropy, the

next important requirement for excellent soft mag-

netic properties is the absence of magnetostriction

in order to minimize magnetoelastic anisotropies. In-

deed, it is a feature of nanocrystalline iron-based

alloys that the phases formed can lead to low or van-

ishing saturation magnetostriction, l

(Fig. 3). This is

responsible for the increase of initial permeability up

to one order of magnitude upon formation of the

nanocrystalline state.

The behavior of l

can be understood from the

balance of the magnetostriction among the a-FeSi

grains and the residual amorphous matrix:

¼ v

FeSi

þð1  v

Þl

ð6Þ

Thus, near-zero magnetostriction in nanocrystal-

line iron-based alloys requires a large crystalline vol-

ume fraction, v

, with negative magnetostriction in

order to compensate for the high positive value of

the amorphous iron-based matrix. This is achieved

either by a high silicon content in the b.c.c. grains

FeSi

E  6  10

6

for a-Fe

) as in the Fe–Cu–

Nb–Si–B system, or if the grains consist of pure

a-iron ðl

FeSi

E  4  10

6

Þ as in Fe–(Cu)–Zr–B.

An important point to stress is that this balance of

local magnetostriction constants to zero really results

in stress-insensitivity of the magnetic properties as,

Figure 2

Initial permeability, m

, and coercivity, H

,of

nanocrystalline Fe

73.5

13.5

(annealed 1h at

540 1C þ4h at 350 1C in a transverse magnetic ﬁeld) vs.

measuring temperature (measuring frequency f ¼50Hz).

The arrows indicate the Curie temperatures of the

residual amorphous matrix (T

¼291 1C) and the b.c.c.

grains (T

FeSi

¼ 607 1C).

Figure 3

Saturation magnetostriction, l

, of Fe–Cu–Nb–Si–B

alloys for the amorphous and nanocrystalline state as a

function of silicon content.

969

Nanocrystalline Materials: Magnetism