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

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since it is essential in order to provide a fair SNR.

However, a higher aspect ratio helps in reducing the

stray ﬁeld lateral extension. Image interpretation

can indeed be greatly simpliﬁed if tip stray ﬁelds may

be approximated by a monopole or well-deﬁned

extended dipole for the operating lift height.

Alternatively, extremely soft, or even paramag-

netic, coatings may be used (Hopkins et al. 1996).

The interaction between the sample and the tip is

then mostly if not solely attractive. A point-like para-

magnetic tip probes the square modulus of the stray

ﬁeld generated by the sample and allows for an accu-

rate stray ﬁeld mapping under specific conditions

(Proksch et al. 1996).

4. Applications of MFM

The need for fast observations of recorded bit pat-

terns fostered the development of MFM. Perpendic-

ular media have been widely studied. Figure 6 shows

an example of dense periodical ﬂux changes in a

CoPtCrO(12nm)/RuCr(4nm)/CoPtCrO(12nm) gran-

ular-type perpendicular recording medium, which

demonstrates a recording performance of 4 Gbit cm

2

(26 Gbit in

2

). Atomically ordered alloys with an L1

type phase, such as FePd or FePt, are known to ex-

hibit an abnormally large uniaxial anisotropy with a

perpendicular to the plane easy axis. Over a wide

thickness range, a stripe pattern develops (Fig. 7(a))

with domains magnetized alternately up and down.

Figure 5

Field distribution in a plane perpendicular to the tip axis and about 50 nm away from the tip apex as measured by

means of electron tomography (courtesy of S. McVitie).

Figure 6

Recorded bits in a granular-type perpendicular

recording medium along tracks with densities ranging

from 50 kfci (kilo ﬂux change per inch) to 400 kfci in

steps of 50 kfci (courtesy of Takao Suzuki and A. Ehsan,

Toyota Technological Institute).

440

Magnetic For ce Microscopy

As the thickness increases, the stripe width ﬁrst

sharply decreases and then slowly increases as the

result of the competition between wall and demag-

netizing ﬁeld energies. In samples grown at room

temperature the chemical ordering is imperfect re-

sulting in a reduced perpendicular anisotropy. So-

called ‘‘weak’’ stripe domains (Fig. 7(b)) as narrow

as B35 nm have been observed for such samples

(Samson et al. 1999) thereby setting an upper limit to

MFM resolution in the favorable case of a periodic

domain pattern. MFM also proves sensitive as clear

initial domain pattern observations in cobalt layers as

thin as three monolayers with the smallest features in

the 400 nm range have been reported.

Patterned magnetic layers are viewed as possible

future high-density recording media. Shape aniso-

tropy now relays growth or chemical ordering-induced

anisotropy. Figure 8 shows a partially demagnetized

array of iron pillars with a large height-to-diameter

aspect ratio, fabricated by scanning tunneling micros-

copy-assisted chemical vapor deposition (Wirth et al.

1999). In this study, the smallest periodicity of par-

ticles was 80 nm corresponding to a data storage

density of B15 Gbit cm

2

(B100 Gbit in

2

). As a pre-

requisite to applications, a precise physical under-

standing of reversal processes in nanometer-scale

particles proves necessary, such as the size threshold

between a pure thermally activated Stoner–Wohlf-

arth-type behavior in weakly interacting particles and

the emergence of more complex nucleation modes.

Because of its ability to handle complex topogra-

phies and of its spatial resolution and sensitivity,

MFM appears, in this context, singular among mag-

netic imaging techniques. Similar arguments prevail

in the observation of magnetic nanowires. One of the

most illustrative observations in 80 nm diameter

Figure 7

MFM images of (a) (fragmented) ‘‘strong’’ and (b) ‘‘weak’’ stripes in chemically ordered FePd alloy thin ﬁlms.

Domain widths as low as B35 nm have been observed in such materials (courtesy of Y. Samson).

Figure 8

MFM image of part of an array of iron pillars grown on

gold. A pulsed ﬁeld of 66 mT has caused reversal of

some of the pillars (color change from bright to black)

(courtesy of S. Wirth).

441

Magnetic For ce Microscopy

nanowires is shown in Fig. 9 (Belliard et al. 1998). In

spite of a rather unpromising geometry, submicro-

meter-size domains are clearly revealed. Image anal-

ysis leads to the picture of axially magnetized wire

segments separated by off-axis magnetized domains,

in agreement with knowledge of the growth proper-

ties of such wires. As the diameter is reduced to

B30 nm, the magnetization distribution remains

axial owing to shape anisotropy, with, because of

magnetic history, the possible occurrence of head-to-

head or tail-to-tail charged walls (Ebels et al. 2000).

If MFM offers unprecedented imaging capabilities

in systems displaying localized charge densities, it

proves much less efﬁcient in the high-resolution

observation of soft magnetic materials, owing to

their very specific ability to screen and/or spread

magnetic charges. In low saturation magnetization

soft magnetic materials such as Permalloy (Ni

the domain wall contrast in micrometer-size elements

proves diffuse, as shown in Fig. 10 (Gomez et al.

1999), whereas large wall distortions are commonly

observed in larger-size elements. Numerical simula-

tions shed light on the underlying physical phenom-

ena (Fig. 11): magnetic charges gather in the soft

material under the inﬂuence of the tip stray ﬁeld.

Their associated demagnetizing ﬁeld opposes the tip

ﬁeld (screening). Tip-induced charges interact with

charges already present in the soft magnetic material

along walls, for instance, thereby modifying the shape

of the wall (Fig. 11(a)). As the tip moves above the

surface of the element, the overall charge pattern

evolves and so does the tip–sample interaction ener-

gy, hence the force or force gradient acting on the tip

(Eqns. (7), (9), and (10)).

If MFM was mapping the charge distribution in-

side the sample without perturbation, then the image

would mirror perfectly the wall symmetry, here a set

of four 901 walls building a perfect ﬂux closure

(Landau–Lifshitz) pattern (Fig. 11(b)). When tip

perturbations are duly accounted for, the MFM

image resembles that of distorted walls (Fig. 11(c))

although it is the result of a complex convolution

process. Simulations performed with realistic tip ﬁeld

distributions properly predict the existence of a

global attractive force between the tip and the soft

element as well as the right order of magnitude for

‘‘wall contrast.’’ Interestingly enough, noticeably sta-

ble images of naturally pinned magnetic structures

such as 3601 walls have been produced in a number of

laboratories (e.g., Cho et al. 1999).

Figure 9

MFM image of a B80 nm diameter magnetic nanowire

showing regions of axial (long sections) and off-axis

(short sections with alternate contrast) magnetization

directions (courtesy of L. Belliard).

Figure 10

MFM images of ﬂux closure domain patterns in Permalloy square elements imaged with a ‘‘low moment’’ tip. From

left to right (mm): 3 3, 2 2, 1 1; thickness 28 nm (courtesy of R. D. Gomez).

442

Magnetic For ce Microscopy

5. Outlook and Perspectives

Further development of MFM needs to include better

magnetic tip calibration and repeatability as well as

environment cells (e.g., temperature and ﬁeld). Mod-

ulation techniques allow the analysis of phenomena

occurring at a time scale much shorter than the res-

onance period of cantilevers (Proksch et al. 1999b).

Besides, a ﬁner interpretation of the mechanics of

force microscopes is also highly desirable: an inde-

pendent monitoring of both the frequency shift and

the cantilever motion amplitude vs. position are key

factors towards a full analysis of dissipation (Proksch

et al. 1999a) and nonlinear oscillations in the high Q

regime (Aime

et al. 1999).

See also: Magnetic Materials: Transmission Electron

Microscopy; Magnets, Soft and Hard: Domains

Bibliography

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Spin accumulation and domain wall magnetoresistance in

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perparamagnetic magnetic force microscope tips. J. Appl.

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Figure 11

Numerical simulations: (a) local perturbation in a

Permalloy platelet (size 1 mm 1 mm) under the action of

a tip representative of standard MFM probes; (b)

simulated MFM image in the ‘‘no perturbation’’ limit;

sample perturbations.

443

Magnetic For ce Microscopy

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J. Miltat and A. Thiaville

Universite

Paris-Sud, Orsay, France

Magnetic Hysteresis

When a magnetic ﬁeld acts on a demagnetized ferro-

magnetic body, the component of the macroscopic

magnetization along the ﬁeld direction increases non-

linearly. In general, the increase takes place with a

low initial slope, a drastic increase of this slope at

intermediate ﬁelds, and a new decrease at higher

ﬁelds up to the technical saturation (total saturation

requires 0 K or applied ﬁelds with strength tending to

inﬁnity). When the ﬁeld is removed, the macroscopic

magnetization decreases along a curve located well

above the initial magnetization curve. For zero ﬁeld

a noticeable magnetization generally remains that is

called remanence or remanent magnetization, M

After reversal of the ﬁeld, the magnetization drops

and vanishes for a reversal ﬁeld known as coercive

ﬁeld, H

. Further increase of the ﬁeld leads to the

saturation. When the ﬁeld is cyclically applied, the

magnetization curve is a hysteresis loop as that

shown in Fig. 1. If the maximum applied ﬁeld is not

high enough to get saturation, the loop is called a

minor loop.

The coercive ﬁeld permits the scaling of the applied

ﬁelds as weak, lower than H

, intermediate, those of

the order of H

, and high ﬁelds, stronger than H

.If

superimposed to a constant ﬁeld, a cyclically varying

ﬁeld is applied, and we obtain a hysteresis loop

around a point of the hysteresis loop, generally

known as a shifted loop. It is important to remark

that the ﬁeld acting on a magnetized sample is the

sum of the externally applied ﬁeld and the demag-

netizing ﬁeld due to the magnetization itself. As the

demagnetizing ﬁeld can be roughly approximated by

–NM, where the demagnetizing factor, N, is geometry

dependent, the shape of the hysteresis loop of any

material depends on its geometry. Customarily, it has

been well established in the textbooks that the ideal

shape of a magnetic sample is either toroidal or a

long narrow bar, both exhibiting negligible demag-

netizing factors. For a more detailed discussion of the

hysteresis in ferromagnetic materials the reader is re-

ferred to the book of Bertotti (1998).

1. Technical and Fundamental Importance of the

Magnetization Curve

Ferromagnetic materials are used for two main tech-

nological applications: (i) as ﬂux multipliers forming

the nucleus of electromagnetic machines, and (ii) as

stores of either energy (magnets) or information (mag-

netic recording). The technical requirements to im-

prove the material performance for both functions are

related to the characteristics of the hysteresis loop and

magnetization process. Flux multiplication requires

high permeability and narrow hysteresis loop or low

Figure 1

(a) Hysteresis loop and initial magnetization curve

showing some characteristic regions: the Rayleigh

region, minor loops at the irreversible region and the

approach to saturation region. (b) Magnetic energy

stored in a ferromagnetic material after being

magnetized up to M

444

Magnetic Hysteresis

coercivity, m

. These properties are characteristic

of the so-called ‘‘soft’’ magnetic materials. Storage

requires high remanence and wide hysteresis loop in

order to prevent demagnetization. Both properties are

characteristic of ‘‘hard’’ magnetic materials.

Even though for all types of applications the higher

the spontaneous magnetization, M

, the better the

performance, the difﬁculty of increasing M

artiﬁ-

cially yields coercivity as the key parameter to be

controlled by the material scientist. In fact, the his-

tory of the magnetic material research is the history

of the progressive increase of the available coercivity

spectrum. The maximum spontaneous magnetization

is that corresponding to 0 K and it is known as sat-

uration magnetization; its value is roughly given by

the atomic magnetic moment, of the order of Bohr

magneton, 10

23

1

, times the number of atoms per

unit volume, typically 10

, that leads to m

of the

order of 1 T. At the beginning of the twentieth cen-

tury, just as in Plato’s time, the harder material,

‘‘hard steels,’’ had a coercivity m

¼10

2

T, only

two orders of magnitude larger than that of the

known softer material, ‘‘puriﬁed iron.’’ At the begin-

ning of the twenty-ﬁrst century, the softer material is

nanocrystalline Fe

with coercivity m

7

T whereas the harder material is nanocrystal-

line Fe

with coercivity m

¼1T.

Seven orders of magnitude separate the coercivity

from the harder to the softer ferromagnetic material

which compositionally differs only in 7 at.% of at-

oms, zirconium for the softer and neodymium for the

harder. This result summarizes the best achievement

of the science of magnetic materials. It has been the

control of the magnetization curve, made possible

from the deep understanding of its governing param-

eter, anisotropy, which has allowed the outstanding

enhancement of the required properties. Anisotropy

can be tailored through both composition and micro-

structure. Since m

M and m

H have the same units it is

possible to compare the hysteresis loops of soft and

hard materials, taking into account that the height of

the loop of soft materials is seven orders of magni-

tude larger than its width, whereas the hysteresis loop

of the best hard materials are square, exhibiting the

same height as width, typically 1 T.

The central characteristics of ferromagnetic mate-

rials, the magnetization curve and the hysteresis loop,

depend on the macroscopic magnetization measured

along the direction of the applied ﬁeld as a function

of the ﬁeld strength. We have seen that magnetization

curve features are of maximum relevance for material

application. From the fundamental point of view, the

problems involved in the physical processes underly-

ing hysteresis, such as relaxation, nonlinearity, meta-

stability, energy dissipation, irreversibility, domain

wall nucleation and propagation, coherent rotation,

or incoherent modes, are the more attractive issues

for basic scientists.

2. Magnetization Processes

Ferromagnetic materials do not generally exhibit

macroscopic magnetization. Even though exchange

interactions tend to align all spins along the same di-

rection, the minimization of the magnetostatic energy

tends to break the spin directional order achieving a

demagnetized macroscopic state. As crystalline ferro-

magnets contain a discrete set of easy directions for

the magnetization, the more common spin conﬁgura-

tion minimizing the sum of exchange, magnetostatic,

and anisotropy energies becomes the domain struc-

ture, as that schematically illustrated by Fig. 2. Do-

mains, within which the spontaneous magnetization,

, is oriented along an easy axis, are separated

by domain walls with thickness d corresponding to

the exchange correlation length. d is of the order of

(A/k)

1/2

where k is the anisotropy constant and A is

the exchange constant A ¼2JS

/a, J being the ex-

change interaction between two adjacent magnetic

moments, S the spin value in h units and a the inter-

atomic distance (see Magnets, Soft and Hard: Do-

mains). In general, A is around 10

11

1

.Sincek

varies from 10

to 10

3

, d ranges from 1 nm to a

few micrometers (Cullity 1972 pp. 290). The energy of

the conﬁguration, anisotropy and exchange, is stored

in the walls with a surface density g ¼(Ak)

1/2

.How-

ever, in sufﬁciently small ferromagnetic particles the

minimum energy corresponds to the single domain

structure. This fact is due to the different geometrical

dependence of the magnetostatic energy that increases

with the sample volume, compared to the wall energy

that increases with the wall area. In the following the

case of multidomain structures will be analyzed.

In multidomain samples each domain is a magnetic

moment. At the demagnetized state the vectorial sum

of the magnetic moments corresponding to each do-

main vanishes. When an external magnetic ﬁeld, H,is

applied along the positive z-direction the Zeeman

energy of the ith domain becomes m

cos y

where v

is the volume of the ith domain and y

the

angle formed by M

of the ith domain with the z-axis,

Figure 2

Schematic representation of two magnetic domains

separated by a domain wall of width d.

445

Magnetic Hysteresis

respectively. Since the Zeeman energy changes from

domain to domain there is an energy gradient through

the walls and, therefore, a pressure exerted on them.

On the other hand, under the torque m

sin y

the

magnetization tends to rotate from the easy axis to the

ﬁeld direction. Under the ﬁeld effect, if the temper-

ature is above 0 K, there is also a tendency to increase

the spontaneous magnetization. This is the so-called

para-effect that is observable in the approach to sat-

uration. Therefore, the changes in the macroscopic

magnetization, DM, can generally be expressed in the

following schematic way

DM ¼ M

cosy

þ M

Dcosy

þ cosy

ð1Þ

The ﬁrst term is known as domain wall motion

contribution, the second corresponds to rotations of

the spontaneous magnetization, and the third de-

scribes the para-effect or increase of the spontaneous

magnetization strength induced by the ﬁeld against

the disordering thermal effect. It has been customarily

assumed that domain wall motion takes place at lower

ﬁelds whereas rotations seemed to require higher ap-

plied ﬁelds. In fact, one of the historically more at-

tractive objectives of researchers in order to increase

coercivity was the production of single domains to

avoid the wall motion contribution to the magnetiza-

tion process. The subsequent development of materi-

als and experimental techniques pointed out that

domain wall motion in high anisotropy samples could

require ﬁeld strengths as large as that required for

magnetization rotation (Becker 1970). However, it

can be accepted that generally domain wall motion is

the main magnetization process at low ﬁelds, whereas

rotations become preponderant for higher ﬁelds. As

stated above, the character of ‘‘low’’ or ‘‘high’’ ﬁelds

should be related to the coercivity and, therefore, de-

pends on the nature of each sample. As shown in

Fig. 1, the initial region of the initial magnetization

curve, the Rayleigh region, corresponds to applied

ﬁelds much smaller than the coercivity and is due to

almost reversible processes. For applied ﬁelds of the

order of coercivity the maximum contribution of ir-

reversible processes takes place. Finally for applied

ﬁelds higher than coercivity that region known as

‘‘approach to saturation’’ is reached.

For a given pressure exerted by the ﬁeld on the wall,

the wall displacement depends on the resistance that

the solid presents to such displacement. Let p be the

pressure acting on a wall initially located at x

, that

for a 1801 domain wall with the ﬁeld parallel to one

domain is of the order of 2m

. Under a pressure p

the wall moves to a new position x at which the pres-

sure exerted by the solid that is due to the position

dependence of the wall energy, dg/dx, exactly coun-

terbalances p. Many models have tried to account for

the spatial dependence of the wall energy density g.

Fluctuations due to modulation of residual stresses,

point defects, dislocations, and grain boundaries

affecting either A or k have been invoked and many

times evidenced as sources of discontinuities in the

domain wall displacement (Ne

el 1978). Nevertheless,

it should be emphasized as the more relevant fact that

the correlation length of the ﬂuctuation, l, must be

scaled to the domain wall width. When lod the wall

energy does not depend on position and therefore the

hindrance to displacement disappears.

This is the case for amorphous materials for which

the structural or topological disorder has a typical

correlation length shorter than d and therefore the

energy of the wall, though high, does not depend

on position. Note that at any relative equilibrium

position x

of the wall at H ¼0 it is veriﬁed

(dg/dx)

¼0. Around x

, g allows a quadratic

Taylor expansion that yields to a reversible linear

displacement for sufﬁciently low ﬁelds. When the

ﬁeld rises leading to an effective pressure p that

overcomes the maximum (dg/dx) the wall undergoes

an irreversible jump and disappears and provides an

order of magnitude for the coercivity. But note that

the derivative dg/dx has only sense for dx4d, there-

fore the maximum dg/dx means the maximum Dg/d

which is of the order of (Ak)

1/2

/(A/k)

1/2

, a ratio that

becomes proportional to k. In other words H

in this

case should increase proportionally to k. Figure 3

illustrates a typical dependence of the wall energy

density with position.

The rotation process has also reversible and irre-

versible steps. Consider a uniaxial anisotropy, of

constant k, that in ﬁrst order can be described by an

energy density term F

¼k(cos

j1/3), where j is

the angle formed by the spontaneous magnetization

and the easy axis, see Fig. 4. As the easy axis forms

an angle y

with the applied ﬁeld the magnetization

Figure 3

Typical dependence of the wall energy density (g) with

position (x).

446

Magnetic Hysteresis

would rotate by an angle j

, from the easy axis

towards the ﬁeld direction, for which the sum of the

anisotropy and Zeeman energy densities vanishes.

Therefore, at the stable equilibrium position, the total

energy should be a minimum and thereby, it is ver-

iﬁed for the ﬁrst derivative



½kðcos



Þm

H cosðy

 f

Þ ¼ 0 ð2Þ

and for the second derivative

½kðcos



Þm

H cosðy

 f

Þ40 ð2aÞ

The component of the magnetization along the

ﬁeld direction M

, that initially at zero ﬁeld was

cos y

, increases with H to M

cos(y

j

). The de-

pendence of M

cos(y

j

) with H can be obtained

from relations (2) and (2a). The solutions are known

as Stoner–Wohlfarth model for coherent rotation

in uniaxial materials (Stoner and Wohlfarth 1948).

Figure 5 summarizes some of the corresponding hys-

teresis loops. It is worth noting that the hysteresis

depends on y

being maximum for the y

¼p case for

which only two stable or metastable states corre-

sponding to j

¼0orj

¼p are allowed and the co-

ercivity corresponds to the so-called anisotropy ﬁeld

¼2k/M

. In contrast, for y

¼p/2 the magneti-

zation curve is reversible and linear with slope M

/2k

for applied ﬁelds smaller than the anisotropy ﬁeld,

i.e., the ﬁeld at which the magnetization ﬁnishes its

rotation reaching the applied ﬁeld direction. For a

random distribution of the easy axes orientation the

coercivity increases with k as well as the ﬁeld required

to reach saturation.

3. Dissipation and Irreversible Processes

One of the more important features of the magnet-

ization process is its irreversibility due to the fact that

for a given ﬁeld there exist many wall positions for

which the energy is a relative minimum. In other

words, for a given applied ﬁeld the macroscopic

magnetization is not a single-valued function of the

applied ﬁeld. For rotation processes also the aniso-

tropy barrier that should be overcome in order to

reverse magnetization yields irreversible jumps and

hysteresis. Note that the magnetization energy or

work supplied in an adiabatic magnetization process

during which, under the action of an increasing

applied ﬁeld, M increases from zero to M

is given

by the integral between zero and M

of m

HdM (see

Fig. 1). If after removing the ﬁeld the magnetization

evolves to the remanent state the expended energy

has not been restored and the process of increasing

and decreasing the applied ﬁeld had the energy cost

indicated in Fig. 1.

During a complete hysteresis loop the energy irre-

versibly lost is that given by the area enclosed by the

loop (see Magnetic Losses). This energy, according to

the M and H scales indicated above, ranges from

3

to 10

1

3

for hard and soft materials,

respectively. The magnetic energy that a magnet can

store is of the order of the energy dissipated during a

complete hysteresis loop. Where does the electro-

magnetic energy go? It is dissipated as heat. Suppose

that the magnetization process of a metallic ferro-

magnet is completely reversible at very low frequency

Figure 4

Rotation of the magnetization from the easy axis under

the inﬂuence of an external magnetic ﬁeld.

Figure 5

Schematic representation of the hysteresis loops of a

uniaxial material according to the Stoner–Wohlfarth

model for coherent rotation. y

is the angle between the

easy axis and the external magnetic ﬁeld.

447

Magnetic Hysteresis

of the applied ﬁeld. As the ﬁeld frequency increases

the electric ﬁeld as well as the eddy currents induced

through the Faraday law rise. The increase of fre-

quency leads to a dissipation of Joule energy as heat

and to the appearance of hysteresis or phase shifts

between ﬁeld and macroscopic magnetization. These

types of losses are known as eddy current losses.

Suppose now that the frequency of the applied ﬁeld is

almost zero. The losses now result only from the

hysteresis of the macroscopic magnetization. In me-

tallic materials the main source of hysteresis losses is

that of the eddy currents.

The walls are locally pinned and suddenly jump as

the ﬁeld varies producing local eddy currents associ-

ated to these processes known as Barkhausen jumps

(Barkhausen 1919). Local or microscopic eddy cur-

rents located around the pinning centre are induced.

Structural relaxation, driven by motion and rear-

rangement of atoms, and magnetization relaxation

through interaction between the spin and lattice sys-

tems also contribute to introduce delays between

magnetization and ﬁeld. All these considerations out-

line the complexity of the magnetization process as a

thermodynamic transformation far from equilibrium,

with time-dependent behavior and irreversible steps

leading to energy dissipation.

4. Anisotropy as the More Important Factor

Determining Hysteresis

From the point of view of the material physicist, the

important conclusion of this overview is the impor-

tance of the anisotropy in the control of coercivity

(see Coercivity Mechanisms). Anisotropy is related to

technical magnetism, magnetization process and hys-

teresis loop just as exchange is to basic magnetism

and magnetic structures. The structural anisotropy is

due to the combined effect of the spin–orbit coupling

and the non-spherical charge distribution around the

magnetic atom. Thus, anisotropy can be increased by

introducing magnetic atoms with large atomic

weight, as rare earth elements. Nowadays, we know

that the average or macroscopic anisotropy can be

controlled from the local or structural anisotropy

through the control of the nanostructure. In partic-

ular, for nanostructured multiphase random aniso-

tropy systems, with the exchange correlation length

larger than the structural ﬂuctuation, the macro-

scopic anisotropy undergoes a noticeable reduction

with respect to the local anisotropy by the smoothing

effect of the exchange energy (Herzer 1989). The im-

portance of the anisotropy on the coercivity is illus-

trated by taking into account that different theories

of coercivity lead to the expression

¼ a



 bM

ð3Þ

where the meaning of the a parameters depend on

the particular theory and magnetization process

(Kronmu

ller et al. 1988, Givord et al. 1991) and b

is a local demagnetizing factor.

5. Single Domains

An important question rises in the case of single do-

mains corresponding to saturated multidomains as

well as in the case of stable single domains; that is,

how can the magnetization reversal take place when

no wall exists over the sample? This is a fundamental

question for the understanding of magnetization re-

versal. Let us suppose that a sample is completely

saturated under an applied ﬁeld along the positive

z-axis that is an easy axis. If after removing the ﬁeld

the macroscopic magnetization decreases, it is be-

cause some domains oriented along the negative

z-axis have been formed. But to nucleate a reversed

seed a ﬁeld of the order of the anisotropy ﬁeld is

required. Therefore the demagnetizing ﬁeld acting on

the sample when the applied ﬁeld is removed should

be higher than the anisotropy ﬁeld. It is well known

that coercivity is generally much lower than that ex-

pected from the value of the anisotropy ﬁeld. This

fact, known as Brown’s paradox, has been interpret-

ed by using two types of main arguments. (i) Exist-

ence of defects yielding local anisotropy effective

constant ﬂuctuations. Such ﬂuctuations would de-

crease at some regions the reversal barrier height. (ii)

The saturated state is not a completely saturated state

but it contains a few very small domains oriented

along the negative z-axis.

These domains, as observed in Fig. 6 in a saturated

Co–Ni multilayer (Gonza

lez et al. 1999), would grow

under the reversal ﬁeld up to a critical size (reached

at the growing ﬁeld) and then propagate through

the whole sample. For this case the nucleation of

reversed domains is not necessary for the magneti-

zation reversal. There are two important ﬁelds, nu-

cleation and propagation ﬁelds, governing the shape

of the hysteresis loop. The concept of nucleation

ﬁeld is generalized to that of growing ﬁeld, when re-

versed domains are frozen. The hysteresis shape is

smoothed if the propagation ﬁeld is higher than the

nucleation one. On the other hand, the hysteresis

loop is rectangular when the propagation ﬁeld is

smaller than the nucleation, which is a condition

of bistability. This property allowed Sixtus and

Tonks (1931) to evidence the contribution of domain

wall motion to the magnetization reversal. A nice

example is shown in the experiment described by

Fig. 7. A positive magnetostriction amorphous wire

keeps closure domains at its end in the saturated

state.

If a uniform ﬁeld larger than the propagation ﬁeld

is applied along the direction opposite to M

and an

extra ﬁeld located at one end is superimposed in such

448

Magnetic Hysteresis

a way that at this end the ﬁeld is higher than the

growing ﬁeld, the wall propagates (Reininger et al.

1993). As in the classic Sixtus and Tonks experiment,

two small pick-up coils, separated by z along the axial

direction z will collect the induced voltage due to the

magnetization reversal with a relative delay, t ¼z/v,

where v is the wall speed during its propagation.

When the length of the wire becomes short enough to

allow overlapping of the closure domains the bista-

bility disappears since the growing ﬁeld decreases

below the propagation ﬁeld. Nowadays theoreticians

and experimentalists proceed the research on mag-

netization reversal mechanisms that form an open

framework for the technical magnetism.

Let us now take into account the case of single

domains. If we focus our interest on small ferromag-

netic particles the nonequilibrium character of the

hysteresis loop can be easily illustrated. Suppose a

set of N uniaxial single domains with their easy axis

perfectly oriented along the z-direction. If this as-

sembly has been saturated under a ﬁeld applied along

the positive z-direction, the system is expected to re-

main saturated just immediately after the ﬁeld was

removed. If we call N

(t) and N



(t) the number of

particles with the magnetization oriented along the

positive or negative z axis respectively, the condition

at t ¼0isN

(0) ¼N. This conﬁguration corresponds

to an equilibrium situation only at T ¼0 K. For

T above 0 K any remanent magnetization at equi-

librium is zero. The free energy (F ¼UTS) minimi-

zation requires minimum energy and maximum

entropy. If we assume the magnetization to lie along

either the positive or negative z-direction the energy

at zero applied ﬁeld is minimum for both cases, since

only anisotropy energy contributes. Therefore, we

have to consider only maximum entropy as equi-

librium condition. The maximum entropy clearly

corresponds to N

¼N



¼N/2. To reach this equi-

librium conﬁguration it takes a time, which is typ-

ically the relaxation time of the system (see Magnetic

Viscosity).

Conceptually speaking it is important to remark

that remanence is always relaxing to demagnetized

state. The hysteresis is due to the fact that the meas-

uring time is much smaller than the relaxation time.

As the relaxation time of a particle of volume v and

anisotropy constant k is of the order of one minute for

kv ¼25K

T, K

being the Boltzmann constant, it is

inferred that a system of particles with kv lower that

25K

T should behave as a paramagnetic (superpara-

magnetic) material without exhibiting any hysteresis

in the ﬁeld dependence of magnetization (Cullity 1972

pp. 410–18). However, by decreasing the temperature

the relaxation time exponentially increases and the

system becomes ferromagnetic exhibiting hysteresis.

The magnetic ﬁeld acts on the magnetization by in-

creasing or decreasing the macroscopic magnetization

in times smaller than the relaxation time, whereas the

thermal motion tends to demagnetize the sample

overcoming the anisotropy barriers and eliminating

the hysteresis phenomena. Therefore, for systems

formed by small particles for which, at reasonable

temperatures, it may be veriﬁed from kv ¼25K

T—

for usual k values of the size of a few nanometers—a

Figure 7

(a)–(d) Magnetic domain structure of an amorphous

wire obtained by magneto-optical Kerr effect showing

the domain wall motion (see text).

Figure 6

Magnetic domain structure of a Co–Ni multilayer

obtained by magneto-optical Kerr effect. The sequence

shows the approach to saturation by increasing the

applied magnetic ﬁeld. Notice the existence in the

saturated state (last image) of domains with the

magnetization opposite to the ﬁeld.

449

Magnetic Hysteresis