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

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For such a system, the magnetization M

(T) van-

ishes at kT

E1.8JS

. For the frustrated antiferro-

magnetic triangular lattice, kT

E1.5JS

is smaller,

reﬂecting closely the change of nearest-neighbor

number z.

For D ¼2, n ¼2 and D ¼2, n ¼3, no phase tran-

sition towards long-range ordering can exist, owing

to the increase of thermally activated spin ﬂuctua-

tions: 7/S

7 ¼0. For n ¼3, the correlation length

diverges exponentially as T-0, according to the re-

lationship:

xðTÞpexpðcJS

=kTÞð8Þ

where c is a constant of the order of 2p. The spin

excitation spectrum of the 2D Heisenberg model is

gapless, and linear or quadratic in q, respectively, for

antiferromagnetic or ferromagnetic couplings. At

wave vectors q smaller than some cut-off value

(roughly corresponding to the relationship

ho(q

)EkT ), the spin dynamics become of the dif-

fusive type. The case D ¼2, n ¼2 is quite pathologic.

For the 2D XY model, there exists a very peculiar

phase transition (the so-called Kosterlitz–Thouless

(KT) transition) below some ﬁnite temperature T

towards a new ‘‘topological’’ order (Kosterlitz and

Thouless 1973). The low-temperature phase is char-

acterized by the absence of long-range ordering to-

gether with a divergence of the spin susceptibility

(w(T) remains inﬁnite down to T ¼0K). The instan-

taneous correlation function decays following a

power-law relationship:

Sp1=r

ZðTÞ

ð9Þ

where Z(T

) ¼kT/2pJS

(indeed deﬁning a line of

critical points with Z(T

) ¼1/4 and T

¼pJS

/2).

The physical origin of this nonconventional behavior

has been attributed to the existence in the low-tem-

perature phase of ‘‘topological’’ vortex–antivortex

pairs melting at the KT transition.

This gives rise to a very special temperature de-

pendence of the correlation length which behaves ex-

ponentially according to the well-known relationship:

xðTÞpexp

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

 1

ð10Þ

where b ¼ p=

ﬃﬃﬃ

.AtTE0K, the low-energy magnetic

excitation spectrum of the 2D XY model is gapless, as

expected for any quasi-isotropic system. It is, more-

over, linear in q at small q (even for the ferromagnetic

case), giving rise to a T

term in the specific heat.

There are no good experimental realizations of the

classical D ¼2, n ¼2 and D ¼2, n ¼3 systems (de

Jongh and Miedema 1974, de Jongh 1990). For the

former case, this situation is mainly ascribed to the

fact that the KT ground state is marginal and easily

destroyed by any small corrective term (interlayer

coupling or axial anisotropy), which should be in-

herent to all real materials. Both tend to induce a

long-range ordering when the temperature becomes

sufﬁciently low.

Experimental investigations exist for the quasi-2D

planar compounds K

CuF

,Rb

CrCl

, BaNi

(PO

)

or BaCo

(AsO

)

(de Jongh 1990), which give some

evidences for the exponential behavior of the corre-

lation length above T

. Concerning the 2D Hei-

senberg model, prototype systems like Rb

MnCl

MnF

exist (de Jongh 1990). Unfortunately, they

all order at ﬁnite temperature owing to non-negligible

interlayer couplings and to the presence of a small

axial single-ion anisotropy term, preventing the ob-

servation of the true 2D behavior.

In the space dimension D ¼1, the strength of spin

ﬂuctuations increases so much that no spontaneous

magnetization can appear down to T ¼0K, even for

the Ising system (Steiner et al. 1976). For D ¼1, n ¼1,

the correlation length obeys the quasi-exponential

relationship:

xðTÞ¼

7lnðtanhðJS

=kTÞÞ7

ð11Þ

where a is the intrachain spin–spin spacing. The ex-

citation spectrum has a spin gap, implying an expo-

nential decay of both the longitudinal susceptibility

w(T) and the specific heat C(T) at low temperature.

For D ¼1, n ¼2 and D ¼1, n ¼3, the correlation

length behaves linearly in T according to the simple

relationship:

xðTÞ¼x

JSð S þ 1Þ=kT ð12Þ

where prefactors x

¼2 and 1, respectively. The exci-

tation spectrum is gapless (_oðqÞpq at small q)for

both systems, a result again in agreement with the

isotropic character of this spin system. As for the 2D

case, there exists a cut-off wave vector q

1

(T)sep-

arating the diffusive regime from the quasi-propagat-

ing spin wave regime. The specific heat C(T) is linear in

T and w(T) remains ﬁnite at low temperature.

One of the best prototypes of the classical Hei-

senberg chain is probably the manganese-based

(S ¼5/2) chain compound ((C

)

)NMnCl

(TMMC), which exhibits magnetic properties in very

quantitative agreement with the theoretical predic-

tions of that model. In particular the relevance of

nonlinear excitations (also known as solitons) in the

thermal properties (Mikeska 1980) has been clearly

demonstrated in TMMC by very comprehensive

NMR and INS studies as functions of temperature

and ﬁeld (Regnault et al. 1982).

Indeed, in real materials there always exists either a

small interchain coupling (quasi-1D system) or a

small interlayer coupling J

(quasi-2D system) which

Bulk Magnetic Materials: Low-dimensional Systems

ﬁnally leads to the occurrence of a phase transition

towards a true long-range ordering at ﬁnite temper-

ature. Qualitatively, the critical temperature T

can

be determined from the implicit relationship:

ðT

ÞEkT

ð13Þ

which means that the phase transition takes place as

soon as the thermal energy is comparable to the ex-

change energy of adjacent correlated segments. For

the quasi-1D isotropic chain, this relationship gives:

E2S

ﬃﬃﬃﬃﬃﬃﬃ

ð14Þ

For the quasi-2D Heisenberg system this implies a

logarithmic dependence on the ratio J

/J:

4pJS

7lnðJ

=JÞ7

ð15Þ

The latter relationship explains why very good pro-

totypes of the ideal 2D Heisenberg and XY systems

are lacking (de Jongh 1990).

3. Effects of Quantum Fluctuations

Quantum ﬂuctuations are expected to become strong-

er as the spin value, S, decreases from inﬁnity (clas-

sical limit) to

(strongly quantum limit). However,

the space dimension, D, the number of adjacent spins,

and the sign of the intrachain or intralayer couplings,

J, also play an important role. Qualitatively, one

ﬁnds that the smaller D , the larger the quantum ef-

fects. Antiferromagnetic interactions are particularly

suitable for the existence of strong quantum ﬂuctu-

ations, the more so if they are frustrating the spin

system. This is especially the case in the antiferro-

magnetic triangular lattice, for the J

–J

antiferro-

magnetic model of the square lattice, or in the J

–J

S ¼

antiferromagnetic chain.

3.1 The Two-dimensional Case

In space dimension D ¼2, for evident topology ar-

guments the predictions are not much different from

the classical case. The thermal behavior is qualita-

tively the same as to some renormalization of co-

efﬁcients or prefactors. Thus, the 2D Ising model

exhibits a phase transition at ﬁnite temperature,

whereas the 2D XY model exhibits again the topo-

logical phase transition. For the 2D Heisenberg mod-

el, the ground and excited states correspond to the

so-called ‘‘spin-liquid’’ state, as far as the frustration

is absent from the problem. The predictions are qual-

itatively the same for both the ferromagnetic and the

antiferromagnetic system. Basically, the quantum an-

tiferromagnetic system exhibits no phase transition at

ﬁnite temperature and remains short-range ordered

down to T ¼0 K. The temperature dependence of the

in-plane correlation length is again exponential, given

by an expression similar to that predicted for the

classical model (as to some renormalization of the

prefactors).

The excitation spectrum at T ¼0 K is gapless and

behaves linearly at small wave vector, _o

Ecq, with

a spin wave velocity, c, only slightly renormalized

with respect to its classical value. The best realiza-

tions of the 2D S ¼

Heisenberg antiferromagnetic

system are the two layered compounds La

CuO

E320 K, JE1500 K) and YBa

410 K, JE1300 K), well known for their very good

high T

superconducting properties appearing upon

hole doping. In both materials, comprehensive INS

studies (Furrer 1998) have allowed an accurate de-

termination of magnetic excitation spectra and a

quantitative comparison with the main predictions

of the S ¼

antiferromagnetic square lattice (in par-

ticular x(T )).

3.2 The One-dimensional Case

In space dimension D ¼1, the strength of quantum

ﬂuctuations is considerably enhanced. The magnetic

behavior depends not only on the sign of the spin–

spin couplings, but also on the parity of the quantity

2S þ1. The case of the antiferromagnetic Heisenberg

chain is the most interesting because in that case the

magnetic properties are found to be very different for

2S þ1 even (S ¼

y)or2S þ1 odd (S ¼1, 2y).

(a) The S ¼

antiferromagnetic chain

The main difference in comparison with the classical

system is seen in the excitation spectrum. As shown

from analytical as well as numerical calculations

(Steiner et al. 1976, Schulz 1986), the spin excitation

spectrum of the S ¼

Heisenberg antiferromagnetic

chain consists of a continuum of triplet (S ¼1) exci-

tations limited on the low-energy side by the thresh-

old energy:

ðq

Þ¼

Jsinð2pq

aÞð16Þ

and on the high-energy side by the threshold energy:

ðq

Þ¼pJsinðpq

aÞð17Þ

The excitation spectrum (unbound spinon contin-

uum (Schulz 1986)) is gapless and the ground state is

not ordered, even at T ¼0 K. The equal-time spin–

spin correlation function behaves according to the

algebraic relationship /S

SpðÞ

lnðnÞ=n, where-

as the magnetic susceptibility at very low temperature

remains ﬁnite and the magnetic specific heat is linear

in T. The most important predictions of the S ¼

Heisenberg antiferromagnetic chain have been quan-

titatively veriﬁed in the two prototype materials

CuCl

d2N(C

) (CPC) (Endo et al. 1974) and

Bulk Magnetic Materials: Low-dimensional Systems

KCuF

(Tennant et al . 1993). Figure 4 shows, for

example, the dispersion relationship along the chain

axis and the wave vector dependence of the energy-

integrated intensity in CPC, both found to be in good

quantitative agreement with the corresponding theo-

retical predictions.

The S ¼

Heisenberg antiferromagnetic chain in

the presence of non-negligible spin–lattice interac-

tions shows a quite different behavior. As shown by

Pytte (1974), the strong 1D spin ﬂuctuations induce a

lattice instability (indeed a dimerization) below a

well-deﬁned temperature, T

, called the spin-Peierls

transition temperature. As a consequence of the

presence of alternating couplings induced by the

dimerization, the ground state of the spin-Peierls sys-

tem is a nonmagnetic singlet (S ¼0) state. It is well

separated from the ﬁrst excited triplet (S ¼1) states

by a gap energy of D

E1.75 kT

, directly related to

through a Bardeen–Cooper–Schrieffer (BCS)-

type relationship, as in phonon-mediated supercon-

ductivity. Under an applied magnetic ﬁeld, the triplet

excited states are linearly split by the Zeeman effect

into three components of energy D

þngm

with n ¼1, 0, 1.

Above a critical ﬁeld H

E0.84D

/gm

, the spin-

Peierls system undergoes a ﬁrst-order phase transi-

tion towards a new incommensurate magnetic phase

which can be viewed as a regular stacking of undis-

torted regions magnetized along the ﬁeld direction

and distorted nonmagnetic regions (deﬁning the so-

called soliton lattice structure). The period, L, of the

soliton lattice is directly controlled by the homoge-

neous magnetization, M, according to MEm

a/L.

This relationship means that each soliton bears a to-

tal spin

. The major predictions of the standard spin-

Peierls model (lattice distortion, singlet ground state,

gap in the excitation, high-ﬁeld incommensurate

phase, etc.) have been quantitatively veriﬁed in sev-

eral good prototype compounds such as TTF–Cu–

BDT (T

E12 K) (Bray et al. 1975) or CuGeO

E14 K) (Hase et al. 1993, Boucher and Regnault

1996). As an example, Fig. 5 shows the temperature

dependence of the static susceptibility in CuGeO

which conﬁrms the spin-Peierls scenario.

Figure 4

Wave vector dependence of the energy-integrated

intensity in CPC. Comparison with the theoretical

prediction for the S ¼

Heisenberg antiferromagnetic

chain.

Figure 5

Temperature dependence of the magnetic susceptibility

along the crystalline directions a, b, and c (chain axis) in

the S ¼

spin-Peierls compound CuGeO

Bulk Magnetic Materials: Low-dimensional Systems

(b) The S ¼1 antiferromagnetic chain

The spin S ¼1 Heisenberg antiferromagnetic chain

displays quite different magnetic properties. As

shown by Haldane (1983) and Afﬂeck (1990), for a

spin S ¼1 the ground state is a nonmagnetic singlet

ground state well separated from the ﬁrst excited tri-

plet states by a large energy gap DE0.41J, where J is

the intrachain coupling constant. Contrary to the

S ¼

Heisenberg antiferromagnetic case, the spin–

spin correlation function behaves exponentially:

SpðÞ

expðna=xÞ

ﬃﬃﬃ

ð18Þ

with a correlation length x which is directly related to

the gap energy D by the relationship xEc/D, with

cE2.7Ja (c being the spin wave velocity for the mod-

el). As a consequence of the spin-gap opening, both

the susceptibility and the magnetic specific heat van-

ish exponentially at low temperature, according to:

wðTÞp

expðD=kTÞ

ﬃﬃﬃﬃ

ð19Þ

and

CðTÞpTexpðD=kTÞð20Þ

In agreement with numerical calculations, the cor-

relation length of the S ¼1 chain remains ﬁnite at

T ¼0K (xE6.5a). Under an applied ﬁeld the Hal-

dane ground state disappears at a critical ﬁeld H

D/gm

, above which a more conventional magnetic

phase is recovered. Experimentally, the nickel-based

organic compound Ni(C

)

ClO

(NENP) is

probably the best and most studied prototype of the

Haldane gap system. In this material, the large Hal-

dane triplet is split into three components by the

presence of an intrinsic orthorhombic single-ion an-

isotropy. Macroscopic as well as microscopic meas-

urements have clearly demonstrated the existence at

low temperature of a Haldane phase and the opening

of three quantum gaps (Renard et al. 1987). Figure 6

shows, for example, the disappearance of the static

magnetic susceptibility along the three main cry-

stallographic directions in NENP, a clear signature of

the existence of both a singlet ground state and a spin

gap at low temperature.

For a larger spin value S, the Haldane gap vanishes

exponentially according to the semiclassical relation

DE2JS exp(pS) (Haldane 1983, Afﬂeck 1990) and

the quantum ﬂuctuations are not strong enough to

prevent the interchain couplings from inducing a 3D

long-range order.

3.3 The Spin-ladder System

At the crossing between chain and plane (see Fig. 1),

the n-leg S ¼

ladder system displays quite interesting

magnetic properties, depending drastically on the

parity of n, the number of legs. As in the purely 1D

case, the most pronounced effects are expected when

the spin couplings are antiferromagnetic, both along

the legs and the rungs. For n odd, the spin system

behaves like the S ¼

chain: gapless magnetic ground

state and power-law decay of spin–spin correlation

functions. For n even, the spin system exhibits a

nonmagnetic singlet ground state separated from the

ﬁrst triplet states by a gap energy D pJ exp(n) and

behaves like a Haldane system (Dagotto and Rice

1996). The largest quantum effect is predicted for the

two-leg (n ¼2) antiferromagnetic ladder with a gap

energy amounting to about 0.53J. Basically, the sin-

glet ground state results from the formation of

strongly coupled S ¼0 dimers along the rungs. The

main interest of the two-leg spin-ladder system is in

the very interesting conducting properties supposed

to appear upon charge doping (e.g., hole doping). As

shown theoretically (Dagotto and Rice 1996), if the

antiferromagnetic couplings along the legs and the

rungs are strong enough to cancel the Coulomb re-

pulsion, the charge carriers tend to be conﬁned to the

rungs, leading to an anisotropic pairing mechanism

of charge carriers of d-wave symmetry, very reminis-

cent of that existing in high T

superconductors.

Experimentally, the n-parity effect has been unam-

biguously observed from susceptibility measurements

in the series Sr

n1

n þ1

(Azuma et al. 1994).

In agreement with theoretical studies, SrCu

two-leg ladder compound) exhibits a susceptibility

vanishing exponentially at low temperature, charac-

teristic of a spin-gap opening. In contrast, Sr

(a three-leg ladder compound) displays a ﬁnite

Figure 6

Temperature dependence of the magnetic susceptibility

along a, b (chain), and c axes in the quasi-1D S ¼1

antiferromagnetic chain compound NENP.

Bulk Magnetic Materials: Low-dimensional Systems

susceptibility at T ¼0 K, characteristic of a gapless

system. The existence of the singlet ground state and

the spin gap has been seen in the insulating ladder

cuprate Sr

from comprehensive NMR

(Kumagai et al. 1997) and INS studies (Eccleston

et al. 1998). This compound is very interesting be-

cause its crystallographic structure (see Fig. 2) can be

described as a stacking of magnetically inert Cu

two-leg ladder layers and CuO

chain layers, sepa-

rated by strontium layers (forming a so-called ‘‘com-

posite’’ structure).

Figure 7 shows a typical INS experimental result

obtained for a single-crystalline sample, demonstrat-

ing clearly the presence in this material of a forbidden

energy window below 33 meV. Interestingly, the

chain subsystem contains a large amount of intrin-

sic quasi-ordered holes ( B0.6/Cu) and plays the role

of charge reservoir. When substituting calcium for

strontium, a part of these holes is released to the

ladders, and the new material (of chemical

formula Sr

14x

) undergoes an insulator–

metal transition at ambient pressure and ﬁnally a

nonconventional superconductivity is induced under

pressure. For calcium content of xE13.6, the super-

conducting transition temperature T

reaches a max-

imum value of 10 K at 30 kbar (Uehara et al. 1996,

Mayaffre et al. 1998).

4. Effects of Impurities

Chemical substitution takes on different aspects

depending on the substituted element. In the most

frequent case, a part of the magnetic ions in the par-

ent material is replaced by magnetic or nonmagnetic

elements of the same chemical nature (e.g., substitu-

tion of zinc for copper or nickel for copper). The

magnetic properties of the doped system will depend

on several factors, among which is the nature of the

ground state (long-range ordered or quantum disor-

dered), the space dimension of the lattice (D ¼1or

D ¼2), and the nature of the impurity itself (valence

state, spin). In some cases, the substitution leads to a

real charge doping and the formerly insulating ma-

terial acquires a metallic character and superconduc-

tivity can even occur.

4.1 Dilution by Magnetic or Nonmagnetic Impurities

The introduction of randomly distributed impurities

breaks the regular network of exchange links cou-

pling the magnetic ions. The effect is particularly

dramatic for a 1D system because the impurity cuts a

chain more or less completely, depending on its spin

value. Obviously, the topology of the parent lattice

plays a large role. The size of the region disturbed

around the impurity will also depend on the strength

of quantum ﬂuctuations. The strongest effects are

again expected for the S ¼

spin-Peierls system or the

S ¼

alternating antiferromagnetic chain, two sys-

tems exhibiting a singlet ground state and a spin gap.

In the gapless quasi-1D or quasi-2D systems, the

main effect of impurities is to reduce the correlation

length (and thus the 3D ordering temperature) and to

bring an additional 1/T contribution to the suscep-

tibility (assuming noninteracting impurities, which is

the case if the impurity density, n

, is small). This is

easily understandable by assuming that the impurity

is ‘‘dressed’’ by a magnetic defect of ﬁnite size.

Within this picture, the correlation length at low

temperature is mainly controlled by the impurity–

impurity distance d

. This distance is directly related

to n

by the geometrical relationship d

Ea/n

1/D

.At

ﬁnite temperature, both the impurity and tempera-

ture effects are at ﬁrst order additive:

1

ðn

; TÞEd

1

þ x

1

int

ðTÞ. Both the magnetic defects

created around each impurity and the intradefect

correlated segments contribute to the spin dynamics.

At low temperature, quasi-propagating damped spin

wave modes exist for wave vectors larger than a cut-

off value q

E2p/d

, whereas at low q a diffusive qua-

si-elastic mode appears. This classical behavior has

been quantitatively observed in the S ¼5/2 Heisenb-

erg antiferromagnetic chain compound TMMC

slightly substituted with copper for manganese

(Dupas and Renard 1978).

The case of strongly quantum antiferromagnetic

chain systems displaying a spin gap D is less intuitive.

Basically, in such systems the correlation length of

spin ﬂuctuations is strongly reduced by the quantum

ﬂuctuations and remains ﬁnite at T ¼0K (xEc/D,

Figure 7

A typical INS result showing the presence of a gap in the

magnetic excitation spectrum of the S ¼

two-leg spin-

ladder dimer-chain material Sr

Bulk Magnetic Materials: Low-dimensional Systems

where c is the spin wave velocity, i.e., typically

xE(37)a depending on the particular system). The

main role of impurities is to suppress the quantum

ﬂuctuations and thus the doped system recovers a

more conventional magnetic behavior, with spin ex-

citations appearing in the gap. In some cases, simul-

taneously with the destruction of the singlet ground

state, a true long-range ordered antiferromagnetic

phase occurs, coexisting with the singlet phase. This

puzzling result (order by disorder!) is attributed to the

fact that around each impurity a magnetic defect is

formed (a soliton in this case) developing a staggered

magnetization S(R

) and extending over a character-

istic length scale of the order of the quantum corre-

lation length x.

Each soliton bears a total spin

which can be de-

tected, for example, by susceptibility or ESR meas-

urements. Such a scenario has been observed in the

S ¼1 Heisenberg antiferromagnetic chain compound

NENP doped with a small amount of copper or

cadmium. The case of the spin-Peierls compound

CuGeO

is even more interesting. For this system, the

substitution of a small amount x of silicon for ger-

manium or y of zinc for copper is enough to consid-

erably reduce T

(DT

E35x for silicon

doping), meaning that the coherence length of the

dimerization associated with the spin-Peierls transi-

tion decreases strongly, even at small doping. Neu-

tron diffraction measurements reveal the presence of

a true long-range ordered antiferromagnetic phase

coexisting with the short-range dimerized phase

(Fukuyama et al. 1996, and references therein). An

explanation has been proposed for such a coexistence

based on the solitonic model (Fukuyama et al. 1996).

Doping by zinc or silicon has qualitatively the same

effect: that of modifying locally the magnetic cou-

plings, either by breaking completely the exchange

links (zinc doping) or by changing the value and/or

the sign of J (silicon doping).

Topological solitons bearing a total spin

are cre-

ated around each impurity. These objects are asso-

ciated with weakly distorted regions extending over a

length scale x and cut the nonmagnetic distorted

chain segments of characteristic length Ba/x (this

value corresponding roughly to the average soliton–

soliton distance). For x large enough, the solitons

overlap and the phase can propagate, giving rise to

the nonzero staggered magnetization detected by

neutron diffraction. Such coexistence is indeed a gen-

eral feature of low-dimensional systems. It has also

been observed in doped spin-ladder materials.

4.2 Charge Doping

In the doping mechanism previously considered, the

substitution between chemical elements having the

same valence states (e.g., Zn

2 þ

for Cu

2 þ

) does not

change the global charge balance. In most cases the

material remains electrically inert. However, there are

systems for which the substitution of elements of dif-

ferent valence states generates charge carriers in the

material. This is, for example, the case when substi-

tuting Sr

2 þ

for La

3 þ

in the ‘‘high T

’’ material

CuO

, for which holes are created in the CuO

planes. In some other cases, the host material con-

tains intrinsically an amount of charge carriers (holes

or electrons) conﬁned to crystallographic entities

playing the role of a charge reservoir. This is the case

in the quasi-2D material YBa

6 þx

(YBCO) or,

as previously seen, in the spin-ladder compound

, for which the charge carriers (indeed

holes) are conﬁned to the CuO chain layers. The

change in oxygen content for the former or the sub-

stitution of calcium for strontium for the latter in-

duces a charge transfer and some holes are released,

respectively, in the CuO

planes and in the Cu

ladders, which become metallic. The superconductiv-

ity can occur either at ambient pressure (as in YBCO)

or under strong pressure (as in the ladder com-

pound). The presence of holes in the CuO

or Cu

layers not only affects the exchange links but also

distorts the surrounding lattice, thus creating mag-

netoelastic defects (usually called polarons). As in the

previous case, the magnetic coherence length should

reﬂect directly the hole–hole distance, Ba=

ﬃﬃﬃﬃﬃ

where n

is the hole density, whereas the spin dy-

namics should be analogous to that of the diluted

system (assuming n

is sufﬁciently small).

The presence of such large-size magnetic defects is

the origin of the strong decrease of T

which is ob-

served in the high T

cuprates upon doping, just be-

fore entering the metallic regime. As shown from

comprehensive neutron scattering investigations in

LSCO and YBCO, the antiferromagnetic long-range

ordering disappears for a hole density as small as

2–2.5% hole Cu

1

. Above this critical value, the pre-

viously gapless excitation spectrum starts to develop

a spin gap directly related to the superconducting

order parameter. In this nontrivial regime, a clear

non-Fermi liquid behavior is observed which origi-

nates from the existence of strong antiferromagnetic

(more or less incommensurate) correlations persisting

in the superconducting state (Furrer 1998).

5. Conclusion and Perspectives

The magnetism in low-dimensional systems often

shows quite nonconventional features resulting in a

large part from an increase in both the thermal and

quantum ﬂuctuations. The general properties of a

given material depend on the subtle balance between

several factors: the spin ﬂuctuations and magnetic

correlations, the (more or less dynamical) charge or-

dering, and the spin–lattice or electron–lattice cou-

plings. Depending on their relative strengths, quite

different ground states can be realized (e.g., insulating

Bulk Magnetic Materials: Low-dimensional Systems

long-range ordered, nonmagnetic singlet, spin-liquid,

ttinger-liquid, and superconducting ground states).

In most cases, the magnetic behavior can be well de-

scribed by considering spin Hamiltonians involving

only the spin degrees of freedom. However, there is a

tendency to reconsider the role played by the orbital

degrees of freedom in the strongly correlated electron

systems. Techniques probing directly both the spin

and orbit correlations (such as spherical neutron

polarimetry or polarized x-ray resonant magnetic dif-

fraction) should bring particularly relevant informa-

tion for the understanding of the roles played by the

spin and orbital ordering.

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L.P. Regnault

CEA, Grenoble, France

Bulk Magnetic Materials: Low-dimensional Systems

Coercivity Mechanisms

Since the early days of the commercial use of per-

manent magnets (PMs) the coercive ﬁeld has been a

topic of theoretical discussion concerning its quanti-

tative interpretation (Becker and Do

ring 1939,

Kersten 1942). With the development of the theory

of micromagnetism in the 1930s and 1940s the basic

concepts became available for a quantitative treat-

ment of magnetization processes and of the hysteresis

loop. At the beginning of the twentieth century the

application of magnetic materials was mainly based

on iron, cobalt, nickel, carbon steels, and natural iron

oxides such as magnetite (Fe

). As a consequence

of the quantum theoretical interpretation of spin or-

der by Heisenberg, Bloch, and Ne

el, combined with

the development of the theory of micromagnetism

by Landau and Lifshitz (1935), Becker and Do

ring

(1939), and Brown (1945), a more systematic tailor-

ing of magnetic materials became possible. The

characterization of the constitution and of the

microstructure allowed an analysis of the microstruc-

ture–property relationships. The development of

high-permeability, high-coercivity, high-remanence,

giant-magnetoresistive, and giant-magnetostrictive

materials combined with suitable electrical and me-

chanical properties has inﬂuenced the whole ﬁeld of

magnetic devices ranging from the energy technique

to high-density recording systems.

The hysteresis loops of these high-quality magnetic

materials are characterized by three main parameters:

coercive ﬁeld, H

, remanence, J

, and initial suscep-

tibility, w

(see Magnetic Hysteresis). In particular,

nanocrystalline alloys have become the leading mate-

rial for high-technology applications. These applica-

tions, the applied experimental techniques, and the

underlying physical effects are summarized in Table 1.

In all these materials the coercive ﬁeld is the basic

property that should be either very large for perma-

nent magnets or extremely small for high-permeability

or sensor materials. For high-density recording sys-

tems H

values of 100–200 kA m

1

are used. The

characteristic parameters of the hysteresis loop de-

pend on the main intrinsic material parameters, such

as the spontaneous polarization, J

¼m

magnetization, m

¼4p 10

7

VsA

1

), the mag-

netocrystalline constants, K

and K

(see Magnetic

Anisotropy), and the magnetostriction, l. Further-

more, the microstructure has a strong inﬂuence on H

and w

. The important role of the anisotropy constant

is demonstrated by Fig. 1 where the dependence of

of different prominent material groups is presented

schematically as a function of K

. Here the coercive

ﬁeld varies over eight orders of magnitude from

1

up to 10

1

. This very large range of

variation is due to the large variations of K

and the

microstructures with length scales varying from nano-

to centimeters. The main mechanisms governing the

coercive ﬁeld are nucleation and domain wall pinning

processes. Sometimes also combinations of rotation

and nucleation processes exist. In the following the

micromagnetic backgrounds of nucleation and pin-

ning in inhomogeneous regions are reviewed.

1. Brown’s Paradox on and the Role of

Microstructures

The ideal coercive ﬁeld of an ellipsoidal, single-domain

uniaxial particle for a uniform rotation process

(Brown 1963) is given by the so-called nucleation ﬁeld:

¼ m

ðN

 N

ÞJ

ð1Þ

According to Eqn. (1), H

is independent of the size

of particles but depends only on the shape via the

Table 1

Nanocrystalline, high-quality magnetic materials.

Application Experimental technique Physical phenomena

High-coercivity–high-remanence

magnets: H

41.5 T; J

41T

Nanocrystalline melt-spinning

Mechanical alloying

High anisotropy Intergrain exchange

coupling

High-permeability–high-remanence

magnets: J

41T; m410

Nanocrystalline melt-spinning

Amorphous crystallization

Low anisotropy Random anisotropy

effect

Giant-magnetostrictive

alloys l410

4

Ion d.c. sputtering Single ion anisotropies of 4f electrons

Giant-magnetoresistive alloys DR/

R450%

Laser ablation Double exchange Magnetic phase

transition

High-density recording: 41Gbcm

2

Multilayers Ion d.c. sputtering Self-

organization of substrates and

particles on substrates

Single-domain granular systems

Perpendicular anisotropy

demagnetization factors N

and N

parallel and per-

pendicular to the easy axis. The coercive ﬁelds pre-

dicted by Eqn. (1) so far have not been realized for

technical materials. Only in the case of elongated iron

whiskers could the theoretical values of the curling

mode be approached (Luborsky and Morelock 1964).

According to Fig. 2, for the most prominent PMs only

20–30% of the theoretically predicted coercive ﬁeld

could be realized for many years. It is now generally

accepted that this discrepancy has to be attributed

to magnetic imperfections that lead locally to a drastic

reduction of the ideal nucleation ﬁeld. The main

sources of this phenomenon, originally known as

Brown’s paradoxon (Brown 1945), are the following:

(i) reduced anisotropy constant at grain surfaces, (ii)

misaligned grains that are exchange coupled with

neighboring grains, and (iii) enhanced local demag-

netization ﬁelds at sharp edges and corners of poly-

hedral grains.

A relationship that has been shown to be very

useful in describing the role of the microstructure and

the temperature dependence of H

is a modiﬁcation

of Eqn. (1) (Kronmu

ller 1987, Kronmu

ller et al.

1988, Sagawa and Hirosawa 1988):

a  N

eff

ð2Þ

where two microstructural parameters, a and N

eff

have been introduced. Here ao1 describes the re-

duction of the ideal crystal ﬁeld 2K

owing to

lattice imperfections and atomic disorder at grain

surfaces. The effective demagnetization factor takes

care of the enhanced stray ﬁelds at edges and corners

of polyhedral grains. Equation (2) not only applies to

nucleation-hardened PMs but also describes H

thin ﬁlms and multilayers and the coercive ﬁeld of the

pinning hardened Sm

-based PMs. The param-

eter a subsumes the above described microstructural

effects and is composed of three subparameters

(Kronmu

ller et al. 1987, Bauer et al. 1996):

a ¼ a

ð3Þ

where a

, a

, and a

describe the effect of a redu-

ced anisotropy constant, of misaligned grains, and

of exchange coupling between grains. An explicit

determination of a is based on solutions of the

micromagnetic equations for special models of micro-

structures.

2. Nucleation Field in Regions of Reduced

Anisotropy Constants

Nucleation ﬁelds are usually derived as the eigenval-

ues of the linearized micromagnetic equations. For

example, the nucleation ﬁeld for uniform rotation has

been obtained for constant material constants J

, K

and A (A ¼exchange constant). Similarly the curling

and the buckling mode have been determined for

constant material parameters (Aharoni 1962). In

magnetically inhomogeneous regions in general all

three material parameters vary spatially. However, K

is much more sensitive with respect to atomic disorder

than J

and A, as known from amorphous alloys. To a

ﬁrst approximation J

and A therefore may be con-

sidered to be constant and only variations of K

are

taken into account. In the case of a planar grain

boundary or a deteriorated surface, the main param-

eters characterizing the magnetic defect are the half

width, r

, and the change, DK, of the anisotropy

Figure 1

Schematic plot of coercive ﬁeld H

vs. crystal anisotropy

for prominent soft, hard, and extremely hard

magnetic materials.

Figure 2

Ratio H

exp

of permanent magnets developed since

the 1950s.

Coercivity Mechanisms

constant. A suitable one-dimensional equation for the

position dependence of K

(z) allowing a quantitative

solution of the nucleation problem is given by

(Kronmu

ller 1987).

ðzÞ¼K

ðNÞDK=ch

ðz=r

Þð4Þ

In the case of a one-dimensional problem, e.g., a pla-

nar grain boundary as shown in Fig. 3, the linearized

micromagnetic equation is then

 2 K

ðNÞ

ðz=r



J

ðH

ext

þ N

eff



j ¼ 0 ð5Þ

where j is the angle between J

and the easy axis. The

solution of Eqn. (5) leads to hypergeometric functions

and the parameter a

follows from the lowest eigen-

value of possible solutions:

¼ 1 

1 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þ

ð6Þ

where d

¼ p

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

A=K

ðNÞ

4 denotes the wall width of

the perfect crystal and d

¼ p

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

A=DK

corresponds to

a ﬁctitious wall width of a material with anisotropy

DK. The parameter a

is presented in Fig. 4 as a

function of d

and for DK as a parameter. From

Eqn. (6), three interesting limiting cases can be de-

rived:

1: 2pr

; a

¼ 1  p

2: 2pr

; a

 DK

3: 2pr

; a

 DK

ð7Þ

In the case where DK ¼K

, a

in the three limits is

¼1p

, a

¼(1/p)(d

), and a

¼0. Inserting

of the second limit into Eqn. (2) gives

¼ H

 N

eff

 N

eff

ð8Þ

where g

¼4

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

denotes the wall energy per unit

area.

3. Nucleation Field of Misaligned Grains

The nucleation ﬁeld is reduced by a factor a

if the

ﬁeld is applied under an angle c

with respect to the

negative c-axis. Stoner and Wohlfarth (1948) have

determined a

for a uniaxial particle and the aniso-

tropy constant K

. An expansion of the Stoner–

Wohlfarth result including the second anisotropy

constant K

is given by (Kronmu

ller et al. 1987)

½ðcosc

2=3

þðsinc

2=3



3=2

 1 þ

þ K

ðtanc

2=3

1 þðtanc

2=3

ð9Þ

The angle j

by which J

is rotated before sponta-

neous nucleation takes place is given by

¼ tan

1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

tanc

þ K

ð10Þ

Figure 3

Nucleation of a reversed domain within a planar

intergranular phase of reduced anisotropy. Left:

perpendicular stripe; right: parallel stripe.

Figure 4

Variation of a

as a function of r

for various DK.

Coercivity Mechanisms